Singular stochastic integral operators
Emiel Lorist Mark Veraar



In this paper we introduce Calderón–Zygmund theory for singular stochastic integrals with operator-valued kernel. In particular, we prove L^p-extrapolation results under a Hörmander condition on the kernel. Sparse domination and sharp weighted bounds are obtained under a Dini condition on the kernel, leading to a stochastic version of the solution to the A_2-conjecture. The results are applied to obtain p-independence and weighted bounds for stochastic maximal L^p-regularity both in the complex and real interpolation scale. As a consequence we obtain several new regularity results for the stochastic heat equation on ^d and smooth and angular domains.




[2020]Primary: 60H15; Secondary: 35B65, 35R60, 42B37, 47D06







[
    N. Korolkova^1 and G. Leuchs^2
    
==================================







§ INTRODUCTION


In the study of stochastic partial differential equations (SPDEs), one often needs sharp regularity results for the linear equations. This together with fixed point arguments can be used to obtain existence, uniqueness and regularity for the solution to nonlinear SPDEs. During the last decades so-called maximal regularity results for SPDEs have been obtained in many papers. We refer to <cit.> for an overview on the subject in the Hilbert space setting. In the L^q-setting sharp regularity results have been obtained in <cit.> by real analysis and PDE methods, and in <cit.> by functional calculus techniques.

In the above approaches one needs to prove sharp regularity estimates for singular stochastic integral operator  of the form

    S_K G(s) := ∫_0^∞ K(s,t) G(t)  W_H(t),      s∈_+,

where G is an adapted process and W_H is a cylindrical Brownian motion (see Section <ref>) and K is a given operator-valued kernel K_+×_+→L(X,Y). An important example of a kernel K is

    K(s,t) = e^-(s-t)A_t<s,

where -A is the generator of an analytic semigroup and Y is either the real interpolation space (X, D(A))_1/2,2, the complex interpolation space [X,D(A)]_1/2 or the fractional domain space D((λ+A^1/2)), where λ∈ρ(-A). This kernel has a singularity of the form K(s,t)≤ C(s-t)^-1/2 for |s-t|<1. The L^p-boundedness of singular stochastic integrals with this kernel leads to stochastic maximal L^p-regularity.

Unlike in the deterministic setting, there is no general theory for the L^p-boundedness of singular stochastic integral operators. The aim of this paper is to provide a version of this theory and to use it to obtain new regularity results for abstract classes of SPDEs and more concrete examples such as the heat equation.



 §.§ Deterministic singular integrals

Before Calderón-Zygmund theory <cit.> was developed, the L^p-boundedness of singular integral operators

    T f := ∫_^d K(s,t) f(t)  t

was considered on a case by case basis. Typically the singularity of the kernel K is of the form |K(s,t)|≤ C(s-t)^-1. Important examples are the Hilbert transform
(for d=1), and the Riesz transforms (for d≥ 2) in which case the integral has to be interpreted as a principal value. Positive kernels are usually easier to deal with as in this case there is absolute convergence and one can apply Schur's lemma (see <cit.>).

In the convolution setting, i.e. K(s,t) = k(s-t),  the Marcinkiewicz–Mihlin multiplier theorem gives simple sufficient conditions on k (the Fourier transform of k) under which T_K is a bounded operator on L^p(^d)  for all p∈ (1, ∞). For detailed expositions on Calderón–Zygmund operators and beyond, we refer to <cit.> and references therein.

The above results have been partially extended to the case where K is scalar valued and f takes values in a Banach space (see the monograph <cit.>). A breakthrough result by <cit.> and <cit.> was that the Hilbert transform is bounded on L^p(;X) with p∈ (1, ∞) if and only if X is a so-called UMD space (see <cit.> for details). Another major breakthrough was given in <cit.>, <cit.> and <cit.>, where the Marcinkiewicz–Mihlin multiplier theorem and Littlewood–Paley decomposition were obtained for the class of UMD spaces.

For a long time operator-valued extensions of the latter results were unavailable outside Hilbert spaces. In <cit.> the notion of R-boundedness was used to obtain a Marcinkiewicz–Mihlin multiplier theorem in the operator-valued setting for d=1. This result was motivated by its applications to maximal L^p-regularity for parabolic PDEs, which were also discussed in <cit.>. In this context the kernel K:^2_+→L(X,D(A)) is given by

    K(s,t) = e^-(s-t)A_t<s,

where -A is the generator of an analytic semigroup. This kernel satisfies K(s,t)≤ C(s-t)^-1 for |t-s|<1. Using Calderón–Zygmund theory one can therefore easily deduce that the L^p-boundedness of the associated singular integral operator for some p_0∈ [1, ∞] implies L^p-boundedness for all p∈ (1, ∞) (see <cit.>). We refer to <cit.> for a detailed discussion on the history of maximal L^p-regularity and to <cit.> for applications to nonlinear PDEs.



 §.§ Singular stochastic integrals

A Calderón–Zygmund theory for stochastic integral operators as in (<ref>) is not available. The behavior of stochastic singular integral operators (<ref>) differs a lot from the deterministic setting. Due to the Itô L^2-isometry the integrals converge absolutely and thus no principal values are needed. As a consequence, in contrast with the deterministic setting, the scalar-valued setting can easily be characterized, see Section <ref>.  In the operator-valued setting cancellation can for example occur in the following form:

    (∫__+K(s,t)x^2_Y  t)^1/2≤ C x_X,       s∈_+,    x∈ X,

where X and Y are Banach spaces. If the kernel is of this form, then using a simple Fubini argument one can check that S_K is L^2-bounded (see Propositions <ref> and <ref><ref>). In particular, this method was used for the kernel in (<ref>) in the classical monograph <cit.>. A sophisticated extension of this type of argument was used in  <cit.>, <cit.> and <cit.> to cover L^p-boundedness in the scale of real interpolation spaces (X, D(A))_θ,p.

The complex interpolation scale is more complicated. In particular, for X = L^p(^d) (<ref>) is often not true. For example it fails for A = -Δ. To obtain L^p-estimates in this case, <cit.> use sharp estimates for stochastic integrals and sophisticated real analysis arguments. Moreover, by using PDE arguments the operator A can be replaced by a second order elliptic operator with coefficients depending on (t,ω,x)∈ [0,∞)×Ω×^d, where some regularity in x is assumed, but only progressive measurability is assumed in (t,ω). By an elaborate trick in <cit.> the estimates were extended to an L^p(L^q)-setting with p≥ q≥ 2. There are many sophisticated variations of the above methods in the literature in which different operators than Δ are considered and equations on different domains D⊆^d are treated (see e.g. <cit.> and references therein).

On the scale of tent spaces stochastic maximal regularity for elliptic operators in divergence form is shown in <cit.>. This is done through extrapolation using off-diagonal estimates, which are substitutes for the classical pointwise kernel estimates of Calderón-–Zygmund theory. See also <cit.> for the more general harmonic analysis framework developed to analyze this scale.

In <cit.> the L^p-boundedness of stochastic singular integrals with kernel (<ref>) was obtained using the boundedness of the H^∞-functional calculus together with the sharp two-sided estimates for stochastic integrals in UMD spaces developed in <cit.>. One of the advantages of this approach is that it can be used for an abstract operator A as long as it has an H^∞-calculus. Secondly, the stochastic integral operator is automatically L^p-bounded for any p∈ (2, ∞). Some geometric restrictions on X are required, but these are fulfilled for L^q, W^s,q, etc. as long as q∈ [2, ∞) (see Section <ref>). In particular, mixed L^p(L^q)-regularity can be obtained for all q∈ [2, ∞) and p∈ (2, ∞), where p=q=2 is allowed as well.
The results of <cit.> have been applied to semilinear equations in <cit.>, to quasilinear equations in <cit.> and to fully nonlinear equations in <cit.>.

Recently, in <cit.> the framework of <cit.> has been extended to cover the case where A depends on time and Ω, as long as D(A(t,ω)) is constant. The method is based on a  reduction to the time and Ω-independent setting and gives a new approach to <cit.>, which additionally includes new optimal space-time regularity estimates and is applicable to a large class of SPDEs.

A large part of the theory of maximal L^p-regularity for deterministic PDEs was developed after the Calderón-Zygmund theory for operator-valued kernels was founded. In the stochastic case such a Calderón–Zygmund theory is not available yet, and our main motivation is to build such a theory and discover its potential for stochastic maximal L^p-regularity (see Subsection <ref>). Our first main theorem in this direction is as follows:


Let X and Y be Banach spaces with type 2 and assume Y is a UMD space. Let K:_+×_+→L(X,Y) be strongly measurable and assume that for every ball B⊂_+ we have the following Hörmander condition

    ∫__+ ∖ BK(s,t)-K(s',t)^2 t^1/2≤ C        s,s'∈1/2B 
    ∫__+ ∖ BK(s,t)-K(s,t')^2 s^1/2≤ C        t,t'∈1/2B

for some constant C>0 independent of B. Fix p∈ [2, ∞) and suppose that the mapping S_K as defined in (<ref>) is bounded from L^p_F(Ω×_+;γ(H,X)) into L^p(Ω×_+;Y). Then for all q∈ (2, ∞) the mapping S_K is bounded from L^q_F(Ω×_+;γ(H,X)) into L^q(Ω×_+;Y).

The above theorem follows from Proposition <ref>, Theorem <ref> and  Theorem <ref> in the homogenous setting (see Section <ref>). In Theorems <ref> and  <ref> we prove a general extrapolation result for singular γ-kernels. In this setting we also obtain the end point estimates L^2→ L^2,∞ and L^∞→. The results are a stochastic version of the classical extrapolation results for Calderón–Zygmund operators (see <cit.> for the scalar case and <cit.> for the vector-valued case).

The conditions (<ref>) and (<ref>) are L^2-variants of what is usually called the Hörmander condition. The L^r-variant for r∈ [1, ∞] also appears in <cit.> in the scalar case and in <cit.> in the vector-valued case, where it was used to extrapolate (deterministic) boundedness of operators from L^p into L^q with 1/p - 1/q = 1/r to other pairs (u,v) satisfying 1<u≤ v<∞ and 1/u - 1/v = 1/r.




 §.§ Weighted boundedness

Next we will state a result on weighted boundedness of stochastic singular integral operators.  For deterministic Calderón-Zygmund operators satisfying the standard conditions, this result was obtained in <cit.>. It settles the so-called A_2-conjecture for standard Calderón-Zygmund operators and states that under standard assumptions on the kernel K one has for all p∈ (1, ∞)

    s ↦∫_^d K(s,t)f(t) t_L^p(^d,w)≤ C_p [w]_A_p^max{1,1/(p-1)}f_L^p(^d,w).

The bound (<ref>) with a non-optimal dependence on the weight characteristic has been known for a much longer time (see <cit.> and references therein).
Originally the A_2-conjecture was formulated for the Beurling–Ahlfors transform in <cit.> where it is shown to imply quasiregularity of certain complex functions. Shortly afterwards it was settled for this operator in <cit.> and subsequently many other operators were treated, which eventually led to <cit.>.

A new proof was obtained in <cit.> where it was shown that any standard Calderón-Zygmund operator can be pointwise dominated by a positive sparse operator. A further extension to Calderón-Zygmund operators satisfying a weaker regularity condition (the so-called Dini condition) was obtained in <cit.>. During the last few years simplified proofs have been obtained by several authors. For our purposes the method in <cit.>, generalized to our setting in <cit.>, is the most suitable and it can be used to obtain the following stochastic version of the A_2-theorem:


Let X and Y be Banach spaces with type 2 and assume Y is a UMD space. Let K:_+×_+→L(X,Y) be strongly measurable and assume that
  
    K(s,t)-K(s',t)   ≤ωs-s'/s-t1/s-t^1/2        s-s'≤1/2s-t,
    K(s,t)-K(s,t')   ≤ωt-t'/s-t1/s-t^1/2        t-t'≤1/2s-t,

where ω:[0,1]→ [0,∞) is increasing and subadditive, ω(0)=0 and

    ∫_0^1ω(r)^2 r/r^1/2 <∞.

Suppose S_K as defined in (<ref>) is bounded from L^p_F(Ω×_+;γ(H,X)) into L^p(Ω×_+;Y) for some p∈ [2, ∞). Then
S_K is bounded from L^q_F(Ω×_+,w;γ(H,X)) into L^q(Ω×_+,w;Y) for all q∈ (2, ∞) and w∈ A_q/2(_+), and the following weighted bound on the operator norm holds

    S_K_L^q→ L^q≲ [w]_A_q/2^max{1/2,1/q-2}.


The above result follows from Theorem <ref> in the homogenous setting (see Section <ref>), which is deduced from a sparse domination result. Furthermore we prove that the above estimate is sharp in terms of the dependence on the weight characteristic. Note that the difference with (<ref>) occurs because the L^p-norm of (<ref>) is equivalent to a certain generalized square function. The conditions (<ref>) and (<ref>) together with the integrability condition on ω are L^2-versions of the so-called Dini condition. The integrability condition on ω holds in particular if ω(t) = Ct^ϵ for some C>0 and ϵ∈ (0,1/2].



 §.§ Consequences for stochastic maximal L^p-regularity


From Theorem <ref> we find that in many instances stochastic maximal L^p-regularity for some p∈ [2, ∞) implies stochastic maximal L^q-regularity for all q∈ (2, ∞) (see Section <ref>). In order to state a particular result here consider the following stochastic evolution equation on a Banach space X:

    du(t) + A u(t)  t  = G(t) d W_H(t),     u(0) = 0.

If -A generates a C_0-semigroup (e^-tA)_t≥ 0 on X, then the solution u is given by

    u(t) = ∫_0^t e^-(t-s)A G(s)  W_H(s).




Assume -A is the generator of a bounded C_0-semigroup on a UMD Banach space X with type 2. Let p∈ [2, ∞) and I = (0,T) with T∈ (0,∞]. Suppose that for all G∈ L^p_F(Ω× (0,T);γ(H,X)) the solution u to (<ref>) satisfies

    A^1/2 u_L^p(Ω× I;X)≲G_L^p_F(Ω× I;γ(H,X)).

Then for all q∈ (2, ∞), w∈ A_q/2(I)
and G∈ L^q_F(Ω× (0,T),w;γ(H,X)) the solution to (<ref>) satisfies

    A^1/2 u_L^q(Ω× I,w;X)≲G_L^q_F(Ω× I,w;γ(H,X)).


The boundedness of the semigroup is only needed if T=∞.
A more general result is contained in Theorem <ref> below. For this we should note that the above L^p-estimate implies sectoriality of angle <π/2 (see <cit.>). Theorem <ref> can be seen as the stochastic analogue of <cit.>. Moreover, the weighted estimates are a stochastic version of <cit.> and <cit.>.

For many differential operators A one can directly apply the results in <cit.> to obtain stochastic maximal L^p-regularity. However, there are numerous situations where this is not the case, for example if:


  *  A does not have a bounded H^∞-calculus;

  *  There is no explicit characterization of D(A^1/2) known;

  *  A(t) and its domain D(A(t)) are time-dependent;

  *  X does not satisfy the R-boundedness condition of <cit.>.

In Corollary <ref> and Remark <ref> we give a situation where <ref> occurs, i.e. we give an example of an operator A without a bounded H^∞-calculus which has stochastic maximal L^p-regularity. In Example <ref>  it seems unknown if <ref>  holds and <ref> seems unavailable as well. In Subsection <ref> we present applications to certain non-autonomous problems where <ref> occurs and in Theorem <ref> we have avoided the geometric restriction mentioned in <ref>.

Another, important novelty is that it is possible to deduce mixed L^p(L^q)-boundedness from L^2(L^2)-boundedness by combining our extrapolation result with the stochastic-deterministic extrapolation result of <cit.>. This reduces the study of the stochastic maximal L^p-regularity problem on L^q to the study of the maximal L^2-regularity problem on L^2 and Green's function estimates (see Remark <ref> and Examples <ref> and <ref>).

The use of temporal A_p/2(_+)-weights in stochastic maximal L^p-regularity is new. In most of the results in <cit.> such weights can also be added without causing major difficulties, but it is very natural to deduce this from extrapolation theory. Moreover with our method we actually obtain a sharp dependence on the A_p/2-characteristic. Weights of the form t^α have already been considered before in <cit.> and can be used to allow rough initial data in stochastic evolution equations. This has become a central tool in deterministic evolution equations (see e.g. <cit.> and references therein). General A_p-weights in parabolic PDEs have been used in <cit.> to derive mixed L^p(L^q)-regularity estimates by Rubio de Francia's weighted extrapolation theorem <cit.>.

In Theorem <ref> and Theorem <ref> one always starts from an L^p-bounded stochastic integral operator. It would be interesting to find general sufficient conditions from which boundedness can be derived. In the deterministic case this can be done using T(1) and T(b)-theorems (see e.g. <cit.> for the vector-valued case).
At least in the Hilbert space setting in the convolution case we obtain a full characterization in Corollary <ref> and Corollary  <ref> assuming a Hörmander and Dini condition, respectively. Finally we mention that it would be interesting to develop a stochastic Calderón–Zygmund theory for noises other than cylindrical Brownian motion.

This paper is organized as follows:


  * In Section <ref> we give some preliminaries on Banach space geometry, γ-radonifying operators, Lorentz spaces, maximal operators and Muckenhoupt weights.

  * In Section <ref> we introduce the stochastic integral operators and establish a connection with γ-integral operators.

  * In Section <ref> we give the definitions of 2-Hörmander kernels and  (ω,2)-Dini kernels and provide some classes of examples.

  * In Section <ref> we prove the extrapolation results for γ-integral operators under a Hörmander condition.

  * In Section <ref> we obtain sparse domination under a Dini condition and use this to prove sharp weighted bounds.

  * In Section <ref> we explain how the results of Sections <ref> and <ref> can be extended to spaces of homogeneous type. Motivated by the application to stochastic integral operators our main example here is the time interval (0,T) with T∈ (0,∞].

  * In Section <ref> we will apply the results of the previous sections to study the p-independence of stochastic maximal L^p-regularity. Here we cover both the time-independent setting and the time-dependent setting using the conditions of Acquistapace and Terreni. Moreover, applications to the (time-dependent) heat equation are given, leading to regularity results in both the complex and real interpolation scale.

  * In Section <ref> we prove a p-independence result on the R-boundedness of stochastic convolutions.

  * Finally in Appendix <ref> we prove some technical kernel estimates.




 §.§ Notation

We denote the Lebesgue measure of a set E ∈B(^d) by E and we often abbreviate the integral of a function f on E as ∫_E f := ∫_E f(s)  s  and the mean as
_E f := 1/E∫ f. A ball with center s and radius r is denoted by B(s,r) and by a cube Q ⊆^d we mean a cube with its sides parallel to the coordinate axes.

For Banach spaces X and Y, L(X,Y) denotes the bounded linear operators from X to Y. If we say that a function f:^d →L(X,Y) is strongly measurable, we mean that K is strongly measurable in the strong operator topology on L(X,Y).

Throughout the paper we write C_a,b,⋯ to denote a constant, which only depends on
the parameters a,b,⋯ and which may change from line to line. By ≲_a,b,⋯ we mean that there is a constant C_a,b,⋯ such that inequality holds and _a,b,⋯ implies ≲_a,b,⋯ and ≳_a,b,⋯.



  §.§.§ Acknowledgements

The authors thank the anonymous referees for careful reading and helpful comments and Petru Cioica-Licht for pointing out a gap in the proof of Proposition <ref><ref>.



§ PRELIMINARIES



 §.§ Banach space geometry

Let X be a Banach space and let (ε_k)_k=1^∞ a sequence of independent Rademacher variables, i.e. uniformly distributed random variables taking values in z ∈:z≤ 1. We say that X has type p ∈ [1,2] is there exists a constant C ≥ 0 such that for all n ∈ and x_1,⋯,x_n ∈ X we have

    ∑_k=1^nε_k x_k_X^p^1/p≤ C  ∑_k=1^n x_k_X^p^1/p.

We say that X has cotype q ∈ [2,∞] if there exists a constant C ≥ 0 such that for all n ∈ and x_1,⋯,x_n ∈ X we have

    ∑_k=1^n x_k_X^q^1/q≤ C  ∑_k=1^nε_k x_k_X^q^1/q.

with the usual modification if q=∞. The least admissible constants C will be denoted by τ_p,X and c_q,X respectively. Note that by randomization equations (<ref>) and (<ref>) imply the same estimates with the Rademacher sequence replaced by a Gaussian sequence, i.e. a sequence of independent standard Gaussian random variables.

All Banach spaces have type 1 and cotype ∞. We say that X has nontrivial type if X has type p ∈ (1,2] and finite cotype if X has cotype q ∈ [2,∞). As example we note that the Lebesgue spaces L^p(^d) and Sobolev spaces W^k,p(^d) have type p ∧ 2 and cotype p ∨ 2. For more details and examples we refer to <cit.>.

We say that a Banach space X has the  property if the martingale difference sequence of any finite martingale in L^p(Ω;X) is unconditional for some (equivalently all) p ∈ (1,∞). That is, if
 there exists a constant C>0 such that for all finite martingales (f_k)_k=1^n in L^p(Ω;X) and scalars |ϵ_k|=1, k=1,⋯,n, we have
 
    ∑_k=1^n ϵ_kdf_k_X^p^1/p≤ C  ∑_k=1^n df_k_X^p^1/p.

 The least admissible constant C in (<ref>) will be denoted by β_p,X.
 Standard examples of Banach spaces with the  property include reflexive L^p-spaces, Lorentz spaces, Sobolev spaces and Besov spaces. For a thorough introduction to the theory of  spaces we refer the reader to <cit.> and <cit.>.




 §.§ γ-radonifying operators
 We recall the definition and some basic properties of γ-radonifying operators, for details we refer to <cit.>.


Let X be a Banach space and H be a Hilbert space. For x ∈ X and e ∈ H we let e ⊗ x be the rank-one operator from H to X given by h ↦h,ex. The γ-radonifying norm of a finite-rank operator of the form T=∑_k=1^n e_k ⊗ x_k, with e_1,⋯,e_n ∈ H orthonormal and x_1,⋯,x_n ∈ X, is defined by

    T_γ(H;X) := ∑_k=1^n γ_kx_k_L^2(Ω;X),

where (γ_k)_k=1^n is a Gaussian sequence on a probability space (Ω,). By the invariance of Gaussians in ^d under orthogonal transformations (see <cit.>), this norm is well-defined. The completion of all finite rank operators from H into X with respect to ·_γ(H;X) is denoted by γ(H,X). Note that in particular γ(H,X)↪L(H,X).

For a measure space (S,μ), we write γ(S;H,X):=γ(L^2(S;H),X) and in particular γ(S;X):=γ(L^2(S),X). Any strongly measurable f S → X, for which f,x^*∈ L^2(S) for all x^* ∈ X^*, defines a bounded linear operator T_f L^2(S) → X by

    T_f φ := ∫_S fφμ,     φ∈ L^2(S),

where the integral is well-defined in the Pettis sense (see <cit.>). If T_f ∈γ(S;X) we say that f represents T_f and write f ∈γ(S;X).

If the Banach space X has type 2 we have the following embedding properties for the γ-spaces, which follow directly from  <cit.>. See <cit.> for the details.

Let X be a Banach space with type 2, H a Hilbert space and (S,μ) a σ-finite measure space. Then we have the embeddings

    L^2(S;γ(H;X))    ↪γ(S;γ(H;X)) ↪γ(L^2(S;H);X)

with both embedding constants bounded by τ_2,X.

Finally, for our γ-version of the A_2-theorem in Section <ref> we will need the following lemma, which follows directly from <cit.>.

Let X be a Banach space with type 2 and let f_1,⋯,f_n ∈γ(S;X) be disjointly supported. Then we have

    ∑_k=1^n f_k_γ(S;X)^2 ≤τ_2,X∑_k=1^nf_k_γ(S;X)^2.







 §.§ Lorentz spaces
 We recall the definition and some elementary properties of Lorentz spaces,
for details we refer to  <cit.>. For a Banach space X a σ-finite measure space (S,μ), p∈ (1,∞) and q∈ [1, ∞] let

    L^p,q(S;X) = (L^1(S;X),L^∞(S;X))_θ,q,     θ = 1p'.

The space L^p,q(S;X) is called the X-valued Lorentz space. An equivalent quasi-norm is given by (see <cit.> and <cit.>):

    f_L^p,q(S;X)    := t↦ t^1/p f^*(t)_L^q(_+,dt/t)
       :=p^1/qt↦ t μ{s∈ S:f(s)_X>t}^1/p_L^q(_+,dt/t),

where f^* denotes the decreasing rearrangement of f_X (see <cit.>). An equivalent norm can be extracted from <cit.>.
For p∈ (1, ∞) one has L^p,p(S;X) = L^p(S;X). In the scalar case L^p,q(S) is a Banach function space.

If p∈ (1, ∞) and q∈ [1, ∞), then the simple functions are dense in L^p,q(S;X). Indeed, this follows from <cit.> and the density of the simple functions in L^r(S;X) for r∈ [1, ∞).

If p∈ (1, ∞) and q∈ [1, ∞] and μ(S) <∞ we have L^p,∞(S) ↪ L^1(S) with

    f_L^1(S)≤μS^1/p'f_L^p,∞(S) ,     f ∈ L^p,∞(S),

which follows directly from the definition and the embedding L^∞(S) ↪ L^1(S) with constant μ(S).

In the next result we extend the γ-Fubini theorem of <cit.> to Lorentz spaces.

Let X be a Banach space and let (S,μ) be a measure space. Then the following assertions hold:


  *  For all p∈ (1, ∞], 
    γ(H,L^p,∞(S;X)) ↪ L^p,∞(S;γ(H,X))


  *   For all p∈ (1, ∞) and q∈ [1, ∞), 
    γ(H,L^p,q(S;X)) = L^p,q(S;γ(H,X))
 isomorphically.



Let E = L^p,q(S) and for a Banach space Y we write E(Y) for the space of strongly measurable functions f:S→ Y such that f_E(Y):=f_Y _E<∞.

We make two preliminary observations.
Since E is a Banach space the triangle inequality in E implies that for all simple functions ξ:Ω→ E,

    ξ_E(L^1(Ω))≤ξ_L^1(Ω;E).

By density this extends to a contractive embedding L^1(Ω;E)↪ E(L^1(Ω)).

The second observation is a certain converse estimate to the above. If p≤ q<∞, we set r=q and if q≤ p<∞ we set r = p+1. Then E is r-concave (see <cit.>). This implies that for all simple functions ξ:S→ L^r(Ω),

    ξ_L^r(Ω;E)≤  C_p,q ξ_E(L^r(Ω)).

By density this can be extended to a contractive embedding E(L^r(Ω))↪ L^r(Ω;E).

Let (h_j)_j=1^n be an orthonormal system in H and let f = ∑_j=1^n h_j⊗ξ_j with ξ_j∈ E(X).
Now setting ξ = ∑_j=1^n γ_j ξ_j_X, where (γ_j)_j=1^n is a Gaussian sequence, we can write

    f_γ(H,E(X))   = ∑_j=1^n γ_j ξ_j_L^2(Ω;E(X)) = ξ_L^2(Ω;E),
    
    f_E(γ(H,X))   = ∑_j=1^n γ_j ξ_j_E(L^2(Ω;X)) = ξ_E(L^2(Ω))

By the Kahane–Khintchine inequalities replacing the L^2(Ω)-norm on the right-hand sides of the above identities with L^r(Ω) with r∈ [1, ∞) leads to an equivalent norm. Taking r=1 we have by (<ref>)  that 
    f_E(γ(H,X))≤ C f_γ(H,E(X)),
 which by density proves γ(H,E(X))↪ E(γ(H,X)) and this proves <ref> and one of the embeddings in <ref>.

To prove <ref> note that by the above with r=p+1  we find by (<ref>)  that 
    f_γ(H,E(X))≤ C_p,q f_E(γ(H,X)).
  Again by density  this gives E(γ(H,X))↪γ(H,E(X)).



Actually in Proposition <ref><ref> the Lorentz space can be replaced by any Banach function space E. Moreover, the extension of <ref> to this setting holds if E is r-concave for some r<∞.

The result of Proposition <ref> can also be extended to quasi-Banach function spaces which are p-convex and q-concave. For the definition of γ(H,Y) for quasi-Banach spaces we refer to <cit.>. In particular, by <cit.> and <cit.> it follows that Proposition <ref><ref> holds for L^p,q(S;X) for all p,q∈ (0,∞).




 §.§ Maximal operators

We define the Hardy–Littlewood maximal operator M for an f ∈ L^1_(^d)  by

    Mf(s) := sup_Q ∋ s_Q f,     s ∈^d,

where the supremum is taken over all cubes Q ⊆^d containing s. For r ∈ (0,∞) and f ∈ L^r_(^d) we define M_rf = M(f^r)^1/r.
These operators satisfy the following bounds:
 Let 0<r<p<∞, then

    M_rf_L^p(^d)   ≤ C_p,r,d f_L^p(^d),       f ∈ L^p(^d),
    M_rf_L^p,∞(^d)   ≤ C_p,r,d f_L^p,∞(^d),       f ∈ L^p,∞(^d),
    M_pf_L^p,∞(^d)   ≤ C_d f_L^p(^d),       f ∈ L^p(^d).


The case r=1 in the first two inequalities and the case p=1 in the third inequality follow for example from Doob's maximal inequalities and a covering argument (see <cit.>). The general cases follow from a simple rescaling argument.


Let X be a Banach space. We define the sharp maximal operator for an f ∈ L^1_(^d;X)  by

    M^#f(s) := sup_Q ∋ s_Q f(t)-_Q f(t') t'_X t,     s ∈^d,

where the supremum is again taken over all cubes Q ⊆^d containing s. Note that  it is immediate from this definition that M^#f ≤ 2 M(f_X), so by Lemma <ref> we have in particular that M^#f ∈ L^p(^d) if f ∈ L^p(^d;X). The converse is also true, which is the content of the next lemma. The proof for the case X = can be found in <cit.>, the general case follows analogously replacing absolute values by norms.


  Let X be a Banach space, 1<r≤ p<∞ and f ∈ L^r(^d;X). Then f ∈ L^p(^d;X) if and only if M^#f ∈ L^p(^d) and
  
    C_p,d^-1 M^#f_L^p(^d)≤f_L^p(^d;X)≤ C_p,d M^#f_L^p(^d).



Lemma <ref> is not valid for p=∞. In this case the space of all f ∈ L^1_(^d;X) such that M^#f ∈ L^∞(^d;X) is strictly larger than L^∞(^d;X). We let (^d;X) be the space of all f ∈ L^1_(^d;X) such that

    f_(^d;X):= sup_Qinf_c ∈ X_Qf(s)-c_X s<∞

where the supremum is taken over all cubes Q ⊆^d. In analogy with  Lemma <ref> we have

    1/2M^#f_L^∞(^d)≤f_(^d;X)≤M^#f_L^∞(^d).

Note that ·_(^d;X) is not a norm, since c _^d_(^d;X) = 0 for any c ∈ X.




 §.§ Muckenhoupt weights
 We recall the basic properties of  Muckenhoupt weights on ^d, for a general overview see <cit.>. Analogous definitions can be given for weights on (0,T) for T ∈ (0,∞].

A weight is a locally integrable function w^d→ (0,∞). For p ∈ [1,∞) and a weight w and a Banach space X we let L^p(^d,w;X) be the subspace of all f ∈ L^0(^d;X) such that

    f_L^p(^d,w;X):= ∫_^df_X^pw^1/p<∞.

and let L^p,∞(^d,w;X) be defined as in Section <ref>.
We will say that a weight w lies in the Muckenhoupt class A_p and write w∈ A_p if it satisfies

    [w]_A_p:=sup_Q_Q w ·_Q w^1-p'^p-1<∞,

where the supremum is taken over all cubes Q⊆^d and the second factor is replaced by (_Q w)^-1 if p=1. Note that w ∈ A_p if and only if w^1-p'∈ A_p' with [w]^1/p_A_p = [w^1-p']_A_p'^1/p' for p ∈ (1,∞).

We will say that a weight w lies in  A_∞ and write w∈ A_∞ if

    [w]_A_∞ := sup_Q ∫_Q M(w _Q)   /∫_Qw   <∞,

where the supremum is taken over all cubes Q⊆^d. Then A_∞  = ⋃_p≥ 1 A_p and for all w ∈ A_p we have

    [w]_A_∞≤ C_d   [w]_A_p,

See e.g. <cit.> for the proof of these facts and a more thorough introduction of the Fuji–Wilson A_∞-characteristic.





§ STOCHASTIC INTEGRAL OPERATORS


For details of the introduced notions in this section we refer to <cit.>.
Let X be a Banach space and H be a Hilbert space. Let (Ω, 𝒜, ) be a probability space with filtration (F_t)_t≥ 0.

Let 𝒲∈L(L^2(_+;H),L^2(Ω)) denote an isonormal mapping (see <cit.>) such that 𝒲 f is F_t-measurable if f=0 on (t, ∞). Define a cylindrical Brownian motion (W_H(t))_t≥ 0 by W_H(t) h := 𝒲 (_[0,t] h).

For f = _(a,b]⊗ h ⊗ξ, where 0≤ a<b<∞ and ξ∈ L^∞(Ω;X) is strongly F_a-measurable, define

    ∫_0^s f(t)  W_H(t) := (W_H(b∧ s) - W_H(a∧ s))h ⊗ξ∈ L^p(Ω;X).

for each s≥ 0. The functions in the linear span of such f are called the finite rank adapted step processes. We extend the definition of the stochastic integral by linearity.

For T∈ (0,∞] and p∈ [1, ∞), we let L^p_F(Ω;γ(0,T;H,X)) denote the closure of all finite rank adapted step processes in L^p(Ω;γ(0,T;H,X)). One has that f∈ L^p_F(Ω;γ(0,T;H,X)) if and only if
f (_[0,s]⊗ h) is strongly F_s-measurable and f∈ L^p(Ω;γ(0,T;H,X)) for all s∈ (0,T) and h∈ H. The following result provides two-sided estimates for the stochastic integral with respect to an H-cylindrical Brownian motion (W_H(t))_t≥ 0.

Let X be a  Banach space, let p∈ (1, ∞) and T∈ (0,∞].
For every adapted finite rank step process f:(0,T)×Ω→L(H,X), one has

    ∫_0^T f(t)  W_H(t)_L^p(Ω;X)_p,Xf_L^p(Ω;γ(0,T;H,X)).

In particular, the mapping f↦∫_0^T f(t)  W_H(t) extends to an isomorphism from L^p_F(Ω;γ(0,T;H,X)) to L^p(Ω;X).




 §.§ Stochastic integral operators

For p∈ [2, ∞), T ∈ (0,∞] and a weight w on (0,T), let L^p_F(Ω× (0,T),w;γ(H,X)) denote the closure of all finite rank adapted step processes in L^p(Ω× (0,T),w;γ(H,X)), where we omit the weight if w ≡ 1. The reason we consider p∈ [2, ∞) will become clear in Subsection <ref>. Although we will not assume type 2 for the moment, it follows from <cit.> that already for very easy kernels K in order to have boundedness of S_K a type 2 condition on Y is necessary.


Let X be a Banach space and Y a UMD Banach space.   Let p∈ [2, ∞), T ∈ (0,∞], w be a weight on (0,T) and let

    K(0,T)× (0,T)→L(X,Y)
 be strongly measurable.  We say that K∈K_W^H(L^p((0,T),w)) if for f∈ L^p_FΩ× (0,T), w;γ(H,X) and a.e. s ∈ (0,T) the mapping t↦ K(s,t) f(t) is in L^p_F(Ω;γ(0,T;H,Y)) and the operator S_K given by

    S_K f(s) := ∫_0^T K(s,t) f(t)  W_H(t),      s ∈ (0,T)

is bounded from L^p_F(Ω× (0,T),w;γ(H,X)) into L^p(Ω× (0,T),w;Y).
We norm K_W^H(L^p((0,T),w)) by

    K_K_W^H(L^p((0,T),w)) := S_K_L^p_F(Ω× (0,T),w;γ(H,X)) → L^p(Ω× (0,T),w;Y).

We omit the weight if w ≡ 1 and we omit the Hilbert space if H =.


We want to study the boundedness properties of S_K. In the next results we will reformulate this problem by reducing to the deterministic setting using square functions (γ-norms in time) and reduce considerations to the case H =.


Let X and Y be a Banach spaces.  Let p∈ [2, ∞), w be a weight on ^d and let 
    K:^d×^d→L(X,Y)
 be strongly measurable. We say that K∈𝒦_γ^H(L^p(^d,w)) (resp. K∈𝒦_γ^H(L^p,∞(^d,w))) if for f ∈ L^p^d,w;γ(H,X) and a.e  s∈^d the mapping t↦ K(s,t) f(t) is in γ(^d;H,Y) and the operator T_K given by

    T_K f(s) := K(s,·) f(·),      s ∈^d

is bounded from L^p(^d,w;γ(H,X)) into L^p(^d,w;γ(^d;H,Y)) (resp. from L^p into L^p,∞). We norm these spaces by

    K_𝒦_γ^H(L^p(^d,w))    := T_K_L^p(^d,w;γ(H,X))→ L^p(^d,w;γ(^d;H,Y)),
    
    K_𝒦_γ^H(L^p,∞(^d,w))    := T_K_L^p(^d,w;γ(H,X))→ L^p,∞(^d,w;γ(^d;H,Y)).

We omit the weight if w ≡ 1 and we omit the Hilbert space if H =. We make the same definitions for ^d replaced by any measure space (S,μ) in the obvious way.


We start by connecting the definitions of stochastic and γ-integral operators.


Let X be a Banach space and Y a UMD Banach space. Let p∈ [2, ∞), T ∈ (0,∞] and let w be a weight on (0,T).
Then 
    𝒦_W^H(L^p((0,T),w)) = 𝒦_γ^H(L^p((0,T),w))
 isomorphically.


The proof is completely straightforward from Theorem <ref>. Indeed if K∈𝒦_γ^H(L^p((0,T),w)), then for f∈ L^p_F(Ω×(0,T),w;γ(H,X)) one has

    S_Kf(s)_L^p(Ω;Y)_p,YT_K f(s)_L^p(Ω;γ(0,T;H,Y)).

Therefore by Fubini's theorem we have

    S_Kf_L^p(Ω×(0,T),w;Y))   _p,YT_K f_L^p(Ω;L^p(w;γ(0,T;H,Y)))
       ≤K_K_γ^H(L^p((0,T),w))f_L^p(Ω;L^p(w;γ(H,X))).
    
        =K_K_γ^H(L^p((0,T),w))f_L^p(Ω× (0,T),w;γ(H,X))

Conversely, taking f independent of Ω, a similar argument yields that K∈𝒦_W^H(L^p((0,T),w)) implies K∈𝒦_γ^H(L^p((0,T),w)).


In the next result we show that one can take H =. The result extends <cit.> where a particular kernel was considered.


Let X and Y be a Banach spaces and (S,μ) a measure space. Assume Y has type 2, let p∈ [2, ∞) and let w be a weight on S.
Then 
    𝒦_γ^H(L^p(S,w))    = 𝒦_γ(L^p(S,w))
    𝒦_γ^H(L^p,∞(S,w))    = 𝒦_γ(L^p,∞(S,w))
isomorphically.


By considering a 1-dimensional subspace of H, we immediately see that ⊆ holds. For the converse let T_K^H and T_K^ be the γ-integral operators on L^pS,w;γ(H,X) and L^pS,w;X respectively.
By Lemma <ref> one has

    T_K^H f(s)_γ(S;H,Y)   ≲_Y T_K^ f(s)_γ(S;γ(H,Y))
      = T_K^ f(s)_γ(H,γ(S,Y))

Taking L^p(S,w)-norms and using Proposition <ref><ref> with p=q we obtain

    T_K^H f_L^p(w;γ(S;H,Y))   ≲_YT_K^ f_L^p(S,w;γ(H,γ(S;Y)))
       _pT^_K f_γ(H,L^p(S,w;γ(S;Y)))
       ≤K_𝒦_γ(L^p(S,w))f_γ(H,L^p(S,w;X))
       _p K_𝒦_γ(L^p(S,w))f_L^p(S,w;γ(H,X)).

The L^p,∞-case follows analogously using
Proposition <ref><ref> instead.




 §.§ Truncations

We will now illustrate a major difference between stochastic and deterministic integral operators. Indeed, we will show that even when the kernel K has a singularity, the “γ-integrals” converge absolutely. In particular, we show that if we truncate the singularity of K, then the operators associated to these truncations converge back to the operator associated to K without any regularity assumptions on K. This is in contrast to the deterministic setting (see <cit.>). For this let X and Y be Banach spaces and suppose that K^d×^d →L(X,Y) is strongly measurable. We define for ε>0

    K_ε(s,t) := K(s,t) _B(0,1/ε)∖ B(0,ε)(s-t),     s,t∈^d.

Let p ∈ [2,∞) and w a weight on ^d. If K_ε∈K_γ(L^p(^d,w)) for all ε>0 we define for f ∈ L^p(^d,w;X) the maximal truncation operator

    T_K^⋆ f(s) :=  sup_ε>0 T_K_εf(s)_γ(^d;Y)     s ∈^d.



Let X and Y Banach spaces and assume that Y has type 2. Let p∈ [2, ∞) and let w be a weight on ^d. Let 
    K^d×^d →L(X,Y)
 be strongly measurable such that K_ε∈K_γ(L^p,∞(^d,w)) for all ε>0. Then for f ∈ L^p(^d;X) we have

    T_K^⋆ f(s) = T_K f(s)_γ(^d;Y),      s ∈^d,

and in particular

    K_𝒦_γ(L^p(^d,w))   =  sup_ε> 0K_ε_𝒦_γ(L^p(^d,w)),
    
    K_𝒦_γ(L^p,∞(^d,w))   = sup_ε> 0K_ε_𝒦_γ(L^p,∞(^d,w)).

Furthermore if K∈𝒦_γ(L^p(^d,w)), then T_K_ε→ T_K in the strong operator topology.


Fix f∈ L^p(^d;X) and s ∈^d. Assume that T_K f(s)_γ(^d;Y)<∞ and take ε>0. Then by domination (see <cit.>)

    T_K_ε f(s)_γ(^d;Y)≤T_K f(s)_γ(^d;Y)

which yields T_K^⋆ f(s) ≤T_K f(s)_γ(^d;Y).


 Conversely assume that T_K^⋆ f(s)<∞. Note that since γ(^d,Y) ↪L(L^2(^d),Y), we have

    ∫_^d | K(s,t) f(t), y^*|^2  t
       ≤sup_ε> 0∫_^d | K_ε(s,t) f(t), y^*|^2  t
    
       ≤sup_ε>0t↦ K_ε(s,t) f(t)_γ(^d;Y)^2 y^*^2<∞.

Therefore, t↦ K(s,t) f(t) is weakly in L^2 and thus T_Kf(s) is a bounded operator from L^2(^d) into Y. Moreover, for all φ∈ L^2(^d) and y^*∈ Y^*, the dominated convergence theorem yields that

    T_Kf(s)φ,y^* = lim_ε→ 0T_K_εf(s)φ,y^*.

Now the γ-Fatou lemma (See <cit.>) yields

    T_Kf(s)_γ(^d;Y)≤lim_ε→ 0T_K_εf(s)_γ(^d;Y) = sup_ε> 0T_K_εf(s)_γ(^d;Y) .

where the equality follows again by domination. This concludes the proof of the equality

    T_K^⋆ f(s) = T_K f(s)_γ(^d;Y).


By taking L^p-norms and using the density of L^p(^d;X) in L^p(^d,w;X) (see <cit.>), we directly obtain

    K_𝒦(L^p(^d,w))   =  T_K^⋆_L^p(^d,w)≤sup_ε> 0K_ε_𝒦(L^p(^d,w)),

and the converse inequality follows from (<ref>). The estimate for L^p,∞ follows analogously. Finally, the strong convergence follows from (<ref>), the dominated convergence theorem and the γ-dominated convergence theorem (see <cit.>).


Next we prove a version of the above result for stochastic integral operators. For this let X and Y be Banach spaces, p ∈ [2,∞) and w a weight on _+. If K_ε∈K_W^H(L^p(_+,w)) for all ε>0 we define  for f ∈ L^p_F(Ω×_+;γ(H,Y)), analogous to T_K^⋆, the operator

    S_K^⋆ f(s) = sup_ε>0 S_K_εf(s)_Y,      s ∈_+.



Let X and Y Banach spaces and assume that Y has  and type 2. Let p∈ [2, ∞) and let w be a weight on _+. Let 
    K_+ ×_+ →L(X,Y)
 be strongly measurable such that K_ε∈K_W^H(L^p(_+,w)) for all ε>0. Then

    S_K^⋆_L^p_F(Ω×_+,w;γ(H,Y)) → L^p(Ω×_+,w)   _psup_ε> 0K_ε_𝒦_W^H(L^p(_+,w))
       _Y,pK_𝒦_W^H(L^p(_+w))

Furthermore if K ∈𝒦_W^H(L^p(_+w)), then  S_K_ε→ S_K in the strong operator topology.


It is clear from Propositions <ref>, <ref> and <ref> that the second and third expression are norm equivalent. Moreover, it is clear that 
    S_K^⋆_L^p_F(Ω×_+,w;γ(H,Y)) → L^p(Ω×_+,w)≥sup_ε>0K_ε_𝒦_W^H(L^p(_+,w)).
 Thus it remains to prove the converse estimate. In order to show this let f∈ L^p_F(Ω×_+,w;γ(H,Y)) and ε∈ (0,1).
Since K∈𝒦_W^H(L^p(_+,w)), by Doob's maximal inequality we can write

    S_K^⋆ f(s)_L^p(Ω)   ≤( sup_ε>0∫_maxs-1/ε,0^maxs-ε,0 K(s,t) f(t)  W_H(t)_Y^p)^1/p
        +( sup_ε>0∫_s+ε^s+1/ε K(s,t) f(t)  W_H(t)_Y^p)^1/p
       ≤4p/p-1S_K f(s)_L^p(Ω;Y).

 Taking L^p(_+,w)-norms the desired estimate follows.

For the strong convergence note that by the proof of Proposition <ref> we have 
    S_K f - S_K_εf_L^p(Ω×_+,w;Y)_p,YT_K f - T_K_ε f_L^p(Ω×_+,w;γ(^d;H,Y)).

Here the right-hand side for fixed ω∈Ω is independent of H by Proposition <ref>, so the strong convergence follows by Proposition <ref> and the dominated convergence theorem.





 §.§ Necessary and sufficient conditions

Before we turn to more involved results in the subsequent sections, we first analyze the boundedness of γ-integral operators in a few special cases. We start with a necessary condition for T_K to be bounded if K is of convolution type.

Let X and Y be Banach spaces, assume that Y has type 2 and let p∈ [2, ∞). Let k:^d →L(X,Y) be strongly measurable and set K(s,t) := k(s-t). If K∈𝒦_γ(L^p,∞(^d)), then for all x ∈ X

    t↦ k(t)x_γ(^d;Y)≤  C_d K_K_γ(L^p,∞(^d))x_X.

The same holds for _+ instead of ^d, where we set K(s,t) = 0 if s ≤ t.



Let r>0, x∈ X and set f = _B(0,2r)⊗ x. Then for all s∈ B(0,r),

    L_r :=t↦ k(t)x_γ(B(0,r);Y)   = t↦ k(s-t)x_γ(B(s,r);Y)
        = T_Kf(s)_γ(B(s,r);Y)≤T_Kf(s)_γ(^d;Y).

Therefore, for any 0<λ<L_r we find that

    λ   ≤λ B(0,r)^-1/p {s∈ B(0,r):Tf(s)_γ(^d;Y)>λ}^1/p
       ≤B(0,r)^-1/p K_K_γ(L^p,∞(^d))f_L^p(^d;X)
       = C_d   K_K_γ(L^p,∞(^d))x_X.

Taking λ = 1/2 L_r, we find that L_r≤ C_d K_K_γ(L^p,∞(^d))x.
Now the proposition follows by letting r→∞ and applying the γ-Fatou lemma (see <cit.>). The proof for _+ is analogous, taking s ∈ (r,2r) instead.


If we replace ^d by (0,T) with T∈ (0,∞)  in Proposition <ref>, we can deduce that
      
    t↦ k(t)x_γ((0,1/2 T);Y)≤  C_d K_K_γ(L^p,∞(0,T))x_X.

      For specific kernels one can  stretch this estimate to the whole interval (0,T) with a constant dependent on T, see <cit.>.


Next we provide some simple sufficient conditions on K for T_K to be bounded using Fubini's theorem and Young's inequality:


Let X and Y be Banach spaces, assume that Y has type 2 and suppose that K^d ×^d →L(X,Y) is strongly measurable. Then the following hold:

  
  *  If there is an A_0>0 such that
  
    s↦ K(s,t)x_L^2(^d;Y)≤  A_0 x_X,      t∈^d,   x∈ X,

then K∈𝒦_γ(L^2(^d)) with K_𝒦_γ(L^2(^d))≤τ_2,Y  A_0.
  
  *  If K(s,t)≤ k(s-t) for some k ∈ L^2(^d), then K∈𝒦_γ(L^p(^d)) for all p ∈ [2,∞) with K_𝒦_γ(L^p(^d))≤τ_2,Yk_L^2(^d).

The same holds for (0,T) with T ∈ (0,∞] instead of ^d, where K(s,t) = 0 if s ≤ t.



For <ref> we have by Lemma <ref> that

    T_K f(s)_γ(^d;Y)≤τ_2,Y(∫_^dK(s,t) f(t)_Y^2  t)^1/2,      s∈^d.

Taking L^2-norms on both sides and applying Fubini's theorem we obtain

    T_K f_L^2(^d;γ(^d;Y))   ≤τ_2,Y∫_^ds↦ K(s,t) f(t)_L^2(^d;Y)^2  t^1/2
       ≤τ_2,Yf_L^2(^d;X).

For <ref> we have by Lemma <ref>

    T_K f(s)_γ(^d;Y)≤τ_2,Y(∫_^d |k(s-t)|^2 f(t)_Y^2  t)^1/2,      s∈^d.

Taking L^p-norms on both sides and applying Young's inequality we obtain

    T_K f_L^p(^d;γ(^d;Y))   ≤τ_2,Y k_L^2(^d)f_L^p(^d;X).

The (0,T) case follows similarly, where we extend K and f by 0 outside (0,T) to apply Young's inequality for <ref>.


If Y is a Hilbert space and K is of convolution type, we can actually characterize the boundedness of T_K, since in this case γ(^d;Y) = L^2(^d;Y). In Corollaries <ref> and <ref> the following result will be improved under regularity conditions on K.
Let X be a Banach space and Y be a Hilbert space. Let k:^d →L(X,Y) be strongly measurable and set K(s,t) := k(s-t). Then the following hold:


  *  K ∈K_γ(L^2(^d)) if and only if t↦ k(t)x_L^2(^d;Y)≲x_X.

  *  If K∈𝒦_γ(L^p,∞(^d)) for some p ∈ [2,∞), then K∈𝒦_γ(L^q(^d)) for all q∈ [2, p).

The same hold for (0,T) with T ∈ (0,∞] instead of ^d, where we set K(s,t) = 0 if s ≤ t.



By <cit.> one has for all t ∈^d that

    s↦ K(s,t)x_L^2(^d;Y) =  s↦ k(s) x_L^2(^d;Y)= s↦ k(s) x_γ(^d;Y),

from which <ref> follows using by Proposition <ref> and <ref><ref>. Part <ref> follows by combining Proposition <ref>, part <ref> and the Marcinkiewicz interpolation theorem (see <cit.>).





 §.§ Scalar kernels

If we allow X to be any Banach space with type 2, but restrict K to be scalar-valued, we can easily characterize the boundedness of T_K if K is of convolution type. This explains why we study the more interesting operator-valued case.

Let X be a Banach space with type 2, let p∈ [2, ∞), let k:^d → be measurable and set K(s,t) := k(s-t). Then T_K is bounded from L^p(^d;X) to  L^p(^d;γ(^d;X)) if and only if k ∈ L^2(^d). Moreover, in this case
K_𝒦_γ(L^p(^d))≤τ_2,Xk_L^2(^d).


Since k is scalar-valued, we have for x ∈ X
  
    s↦ k(s) x_γ(^d;X) = x_Xk_L^2(^d).

  Therefore the result follows from Proposition <ref> and Proposition <ref><ref>.


 In the scalar case, i.e. X = Y =, the L^p-boundedness of T_K can also be well-understood from existing theory for non-convolution kernels. Indeed, in this case K∈𝒦_γ(L^p(^d)) is equivalent to

    ∫_^d(∫_^d |K(s,t)|^2 g(t)  t)^p/2 s ≤ C^p g^p_L^p/2(^d),

where we have set g(t) = |f(t)|^2. The validity of the above estimate is completely characterized by the optimality of Schur's lemma (see <cit.>) applied to the positive kernel |K(s,t)|^2. Moreover, in this case T_K is also bounded in the vector-valued setting where X = Y has type 2, since by Lemma <ref>

    T_K f_L^p(^d;γ(^d;X))^p ≤τ_2,X^p ∫_^d(∫_^d |K(s,t)|^2 g(t)  t)^p/2 s,

where g(t) = f(t)^2_X.
Conversely, by considering a one-dimensional subspace of X, one obtains that (<ref>) is also necessary.

 


  * Let d=1 and K(s,t) = 1/(s+t)^1/2_s,t>0. Then by <cit.> we know that K∈𝒦_γ(L^p()) if and only if p∈ (2, ∞). More generally for 1≤ j ≤ d set
    
    K_js,t := (s_j+t_j)^1/2/s+t^(d+1)/2_s_j,t_j>0,      s,t ∈^d.

Then we know by <cit.> that K_j∈𝒦_γ(L^p(^d)) if and only if p∈ (2, ∞).

  *  If K(s,t) = 1/|s-t|^1/2, then for all p∈ [2, ∞), we have K∉𝒦_γ(L^p()), which is immediate from Proposition <ref>.



Example <ref><ref> can be seen as the analog of the Hilbert transform. It is not bounded for any p ∈ [2,∞) due to the lack of cancellation in the stochastic, scalar-valued setting. This further exemplifies the difference between the deterministic and the stochastic theories.


  The scalar case also shows why we only consider p∈ [2, ∞). Boundedness for p<2 holds if and only if K ≡ 0 (see <cit.>). This also holds for the operator-valued case since L^p-boundedness with p<2 would imply that
K(t,s)x, y^* = 0 a.e. for all x∈ X and y^*∈ Y^*. By strong measurability of (t,s)↦ K(t,s)x this implies that for all x∈ X, we have K(t,s)x ≡ 0. Thus by the density of L^p(^d)⊗ X in L^p(^d;X), we find that K(s,t) f(t) = 0.




§ SINGULAR Y-KERNELS OF HÖRMANDER AND DINI TYPE

Motivated by the connection between stochastic integral operators and γ-integral operators proven in Proposition <ref> and Proposition <ref>, we will now start the systematic study of the K_γ-classes for more involved kernels than those treated in Subsection <ref>. In particular, we will develop a γ-version of the Calderón–Zygmund theory for (deterministic) singular integral operators. This will first be done on ^d and afterwards in Section <ref> we will point out how our arguments carry over to the more general setting of spaces of homogeneous type, which for example includes the (0,T)-case for T ∈ (0,∞].

Let us first define our assumptions on the γ-kernels K:

  Let X,Y be a Banach spaces and let K^d ×^d →L(X,Y) be strongly measurable.
  
    
  * We say that K is a 2-Hörmander kernel if for every ball B⊆^d we have
  
    ∫_^d ∖ BK(s,t)-K(s',t)^2 t^1/2≤ C        s,s'∈1/2B 
    ∫_^d ∖ BK(s,t)-K(s,t')^2 s^1/2≤ C        t,t'∈1/2B

  for some constant C>0 independent of B. The least admissible C will be denoted by K_2.
  
  * We say that K is an (ω,2)-Dini kernel if
  
    K(s,t)-K(s',t)   ≤ωs-s'/s-t1/s-t^d/2        s-s'≤1/2s-t,
    K(s,t)-K(s,t')   ≤ωt-t'/s-t1/s-t^d/2        t-t'≤1/2s-t,

  where ω:[0,1]→ [0,∞) is increasing, subadditive, ω(0)=0 and
  
    K_ω2:=∫_0^1ω(r)^2 r/r^1/2 <∞.

  
  * We say that K is an (ϵ,2)-standard kernel if K is an (ω,2)-Dini kernel with ω(r) = C r^ϵ for some ϵ∈ (0,1] and we set  
    K_ϵ2:= K_ω2.

  
  



  If K is of convolution type, i.e. if K(s,t) = k(s-t) for some k^d →L(X,Y), the Hörmander and Dini conditions in Definition <ref> can be reformulated using a change of variables. Indeed,
(<ref>) and (<ref>) both simplify to

    ∫_1/2s≥tk(s-t)-k(s)^2 s^1/2≤ C       t ∈^d

and (<ref>) and (<ref>) both simplify to

    k(s-t)-k(s)≤ωt/s1/s^d/2    t≤1/2s.

An L^r-variant of (<ref>) already appeared in <cit.>.


 By definition an (ϵ,2)-standard kernel is also an (ω,2)-Dini kernel. As in the deterministic setting an  (ω,2)-Dini kernel is also a 2-Hörmander kernel. The proof is a straightforward adaptation of the proof in the deterministic setting. For the convenience of the reader we include the details.

   Let X,Y be Banach spaces and suppose that K^d ×^d →L(X,Y) is an (ω,2)-Dini kernel. Then K is a 2-Hörmander kernel with
   
    K_2≤ C_d K_ω2.



 We will only show (<ref>), as (<ref>)
 follows analogously.
  Let B=B(s,r)⊆^d be a ball and take s^1,s^2 ∈1/2B. Then  s-s^j≤1/2r ≤1/2s-t for any t ∈^d ∖ B, so
  
    ∫_^d ∖ B   K(s^1,t)-K(s^2,t)^2 t^1/2
       ≤∑_j=1^2 ∫_^d ∖ BK(s,t)-K(s^j,t)^2 t^1/2
       ≤∑_j=1^2 ∫_s-t>rωs-s^j/s-t^21/s-t^d t ^1/2
       ≤ C_d  ∑_k=0^∞ω(2^-k-1)^2 ∫_2^kr<s-t≤ r2^k+1 t/s-t^d^1/2
       ≤ C_d  ∑_k=0^∞ω(2^-k-1)^2 ∫_B(0,r2^k+1)1/(2^kr)^d t^1/2
       ≤ C_d K_ω2.



If K is differentiable, we can check the standard kernel conditions in terms of the derivatives of the kernel.

  Let X and Y be a Banach spaces and let
  
    K ∈ C^1^d×^d∖(s,s):s∈^d;L(X,Y).
 Suppose that there is a constant A_0>0 such that
  
    ∂^α_s K(s,t)   ≤ A_0·s-t^-d/2 -1    α=1,   s ≠ t,
    ∂^α_t K(s,t)   ≤ A_0·s-t^-d/2 -1    α= 1,   s ≠ t.

Then K is a (1,2)-standard kernel with K_12≤ C_d   A_0.



We will prove (<ref>), the proof of (<ref>) is analogous. Take  s,s',t ∈^d such that 0<s-s'≤1/2s-t. Then
we have for all λ∈ [0,1]

    s-t-λ (s-s')   ≥s-t -λ(s-s')≥s-t -s-s'≥1/2s-t.

Therefore using the fundamental theorem of calculus we obtain

    K(s,t)-K(s',t)   = ∫_0^1 /λ  K(s-λ(s-s'),t)λ
       ≤∑_j=1^d∫_0^1 ∂_j K(s-λ(s-s'),t)·(s_j-s'_j)λ
       ≤ A_0 ∑_j=1^d ∫_0^1 s-s'/s-t - λ(s-s')^d/2+1λ
       ≤ C_d A_0 s-s'/s-t1/s-t^d/2

proving the lemma.


If K is of convolution type, a sufficient condition for (<ref>), (<ref>), (<ref>) and (<ref>) can also be formulated in terms of smoothness of the Fourier transform of k. For the usual Hörmander and Dini kernels this is classical. The L^2-variants (or even the L^r-variants) in Definition <ref> can be treated by similar methods (see e.g. <cit.>).

We end this section with a sufficient condition for the standard kernel conditions in terms of fractional smoothness on .

Let Φ_+→L(X,Y) be strongly measurable and suppose there exists a constant A_0>0 and an ϵ∈ (0,12) such that

    Φ(s)≤ A_0  s^-1/2-ϵ,      s>0.

Let k→L(X,Y) be defined by

    k(s) =
    1/Γ(ϵ)∫_0^s (s-r)^ϵ-1Φ(r)  r       s> 0
     0       s ≤ 0

Then K(s,t):= k(s-t) is an (ϵ,2)-standard kernel.


Let s>0 and assume t ∈ [s, 3/2s]. By (<ref>) it suffices to show

    k(s) - k(t)≤ A_0  C_ϵ(t-s)^ϵ/s^ϵ1/s^1/2.

To show this note that

    Γ(ϵ)     k(s) - k(t)
       ≤∫_s^t (t-r)^ϵ-1Φ(r) r + ∫_0^s ((s-r)^ϵ-1 - (t-r)^ϵ-1) Φ(r) r
    
       ≤ A_0 ∫_s^t (t-r)^ϵ-1 r^-ϵ-1/2 r_[15pt]A + A_0  ∫_0^s ((s-r)^ϵ-1 - (t-r)^ϵ-1) r^-ϵ-1/2 r_[15pt]B.

For [15pt]A note that

    [15pt]A≤ s^-ϵ-1/2∫_s^t (t-r)^ϵ-1 r = ϵ^-1(t-s)^ϵ/s^ϵ1/s^1/2.

For [15pt]B we write [15pt]B=[20pt]B1+[20pt]B2 where we have split the integral into parts over (0,s/2) and (s/2,s). For [20pt]B1 we can write

    [20pt]B1     = 1/1-ϵ∫_0^s/2∫_s-r^t-r x^ϵ-2 x   r^-ϵ-1/2 r
    
       ≤1/1-ϵ∫_0^s/2 (t-s) (s-r)^ϵ-2 r^-ϵ-1/2 r
    
       ≤(t-s) (s/2)^ϵ-2/1-ϵ∫_0^s/2 r^-ϵ-1/2 r
    
       ≤(t-s) (s/2)^ϵ-2/(1-ϵ)(1/2-ϵ) (s/2)^1/2-ϵ = 2√(2)/(1-ϵ)(1/2-ϵ)t-s/s1/s^1/2

where we used ϵ<1/2.
Finally,  using t≥ s, we obtain

    [20pt]B2   ≤ (s/2)^-ϵ-1/2∫_s/2^s ((s-r)^ϵ-1 - (t-r)^ϵ-1)   r
    
        = ϵ^-1 (s/2)^-ϵ-1/2((t-s)^ϵ + (s/2)^ϵ - (t- s/2)^ϵ)
    
       ≤ϵ^-1 2^ϵ+1/2(t-s)^ϵ/s^ϵ1/s^1/2,

which implies the required estimate.







§ EXTRAPOLATION FOR Y-INTEGRAL OPERATORS

In this section we will prove the first results regarding the extrapolation of the L^p-boundedness of an γ-integral operator T_K to the L^q-boundedness of T_K for all q ∈ (2,∞) under a 2-Hörmander assumption on K. We will also obtain a weak L^2-endpoint and a -endpoint result.



 §.§ Extrapolation for 2<q<p

Let us start our analysis with an extrapolation result downwards. We will show that if K ∈K_γ(L^p,∞(^d)) satisfies the 2-Hörmander condition, then also K ∈K_γ(L^q(^d)) for all q ∈ (2,p) and K ∈K_γ(L^2,∞(^d)). For this we will adapt the Calderón-Zygmund decomposition technique for singular integral operators to the γ-case. Our main tool will be the following L^2-Calderón–Zygmund decomposition.


  Let X be a Banach space. For every f ∈ L^2(^d;X) and λ >0 there exists a decomposition f = g +b with
  
    g_L^∞(^d;X)≤ 2^d/2λ ,        g_L^2(^d;X)≤f_L^2(^d;X)and b = ∑_j b_j with    b_j ⊆ Q_j           ∑_jQ_j≤λ^-2f_L^2(^d;X)^2
       b_j_L^2(^d;X)^2 ≤ 2^d+2λ^2Q_j,          ∑_kb_j_L^2(^d;X)^2 ≤ 2^d+2f_L^2(^d;X)^2

  for disjoint cubes Q_j ⊆^d.

For the proof in the case X= we refer to <cit.>, where the more general L^q-Calderón–Zygmund decomposition for any q ∈ [1,∞) is shown. The proof carries over verbatim to the vector-valued setting, replacing absolute values by norms.

Note that in the deterministic setting the functions b_j in a Calderón–Zygmund decomposition are usually also taken such that ∫_Q_kb_j = 0, but we will not be able to use this property for γ-integral operators. Instead we use the L^2-Calderón–Zygmund decomposition in a way that is inspired by <cit.>, which builds upon ideas developed in <cit.>.


  Let X and Y be Banach spaces with type 2, let p ∈ [2,∞) and suppose that K ∈K_γ(L^p,∞(^d)) satisfies the 2-Hörmander condition. Then
 
 
  *  K ∈K_γ(L^q(^d)) for all q ∈ (2,p) with
 
    K_K_γ(L^q(^d))   ≤ C_p,q,d τ_2,Xτ_2,YK_K_γ(L^p,∞(^d)) + τ_2,YK_2.

  *   K ∈K_γ(L^2,∞(^d)) with

    K_K_γ(L^2,∞(^d))   ≤ C_p,d τ_2,Xτ_2,YK_K_γ(L^p,∞(^d)) + τ_2,YK_2.

  



 It suffices to show <ref>, as  <ref> then follows directly  from the Marcin­kie­wicz interpolation theorem, see e.g. <cit.>.

 Let f ∈ L^2(^d;X)∩ L^p(^d;X) be compactly supported, λ>0 and set A_0:= K_K_γ(L^p,∞(^d)). Let f = g+b be the L^2-Calderón–Zygmund decomposition of f at level μλ for some μ>0 to be chosen later. Then we have
  
    g_L^p(^d;X)^p≤g_L^∞(^d;X)^p-2g_L^2(^d;X)^2≤2^d/2μλ^p-2f_L^2(^d;X)^2,

  so in particular g ∈ L^p(^d;X). It follows that b =f-g ∈ L^p(^d;X), and thus
  
    T_Kf = T_Kg + T_Kb

  is well-defined.

To estimate the L^2,∞(^d;γ(^d;Y))-norm of T_Kf we need to analyze the size of T_Kf_γ(^d;Y)>λ. We split as follows:
  
    T_Kf_γ(^d;Y)>λ≤T_Kg_γ(^d;Y)>λ2 + T_Kb_γ(^d;Y)>λ2.

  For the term with the “good” part g  we have by our assumption on T_K and (<ref>) that

    T_Kg_γ(^d;Y)>λ/2   ≤A_0^p/(λ/2)^pg^p_L^p(^d;X)
       ≤ C_p,d A_0^p μ^p-2 f_L^2(^d;X)^2/λ^2


For the term with the “bad” part b, let Q_j be the cube corresponding to b_j with center s_j and diameter 2r_j. Set B_j:=B(s_j,r_j), then Q_j ⊆ B_j and B_j≤ C_d Q_j. Set B_j' = 4B_j and O := ⋃_j B_j'.

 for our estimates we will define some auxiliary operators. For r>0 define

    ψ_r(s,t) := 1/B(t,r)^1/2_B(t,r)(s),      s,t ∈^d.

Note that ψ_r(·,t)_L^2(^d)=1 for all t ∈^d and

    sup_t ∈ B_jψ_r_j(s,t)= 1/B_j^1/2_2B_j(s),      s ∈^d

Let S_j L^2(^d;X) → L^2(^d;γ(^d;X)) be the γ-integral operator given by

    S_jh(s) :=  ψ_r_j(s,·)h(·) ,     s∈^d,    h ∈ L^2(^d;X)

which is bounded by Proposition <ref><ref>.
We claim that ∑_j S_jb_j converges in L^p(^d;γ(^d;X)). To prove this we first estimate for fixed j and a.e. s ∈^d

    S_jb_j(s)_γ(^d;X)^2
        ≤τ_2,X^2 ∫_B_jψ_r_j(s,t)^2 b_j(t)_X^2 t
       ≤τ_2,X^2  b_j_L^2(^d;X)^2sup_t ∈ B_jψ_r_j(s,t)^2
     
       ≤ C_d τ_2,X^2   (μλ)^2 _2B_j(s)

using Lemma <ref>, the norm estimate of b_j in terms of Q_j and (<ref>).
Thus for φ∈ L^p'(^d) positive we have

    S_jb_j_γ(^d;X),φ   ≤ C_d τ_2,X μλ∫_Q_j_2 B_jφ(s) s  t 
       ≤ C_d τ_2,X μλ _Q_j,Mφ.

So summing over j we get, using the boundedness of the maximal operator M as in Lemma <ref> and the fact that the Q_j's are disjoint, that

    ∑_j S_jb_j_L^p(^d;γ(^d;X))   ≤∑_j S_jb_j_γ(^d;X)_L^p(^d)
       = sup_φ_L^p'(^d)≤ 1∑_j S_jb_j_γ(^d;X),φ
       ≤ C_d τ_2,X μλ ∑_j _Q_j_L^p(^d)sup_φ_L^p'(^d)≤ 1Mφ_L^p'(^d)
       ≤ C_p,d τ_2,X μλ ∑_jQ_j^1/p.

Since ∑_jQ_j≤ (μλ)^-2f_L^2(^d;X)^2 it follows that ∑_j S_jb_j converges in L^p(^d;γ(^d;X)) as claimed and in particular we have

    ∑_j S_jb_j_L^p(^d;γ(^d;X))≤ C_p,d τ_2,X  (μλ)^1-2/pf_L^2(^d;X)^2/p.


Next set

    ψ(s,t): = ∑_j ψ_r_j(s,t) _Q_j(t)

and define for h ∈ L^2(^d;X) and a.e. s ∈^d

    T_ψ h(s) := (t,t') ↦ K(s,t)ψ(t',t)h(t).

Note that since T_K h(s) ∈γ(^d;Y) by assumption, we also have T_ψ h(s) ∈L(L^2(^d×^d),Y). Moreover since ψ(·,t)_L^2(^d)=1 we have

    (t,t') ↦K(s,t)ψ(t',t)h(t),y^*_L^2(^d×^d) = t ↦K(s,t)h(t),y^*_L^2(^d)

for every y^* ∈ Y^*. Thus by domination (see <cit.>)  it follows that T_ψ h(s) ∈γ(^d×^d;Y) with

    T_Kh(s)_γ(^d;Y)   =T_ψ h(s)_γ(^d×^d;Y).

Finally let

    T_Kγ(^d;L^p(^d;X)) →γ(^d;L^p,∞(^d;γ(^d;Y)))

 be the canonical extension of T_K, which is trivially bounded with norm A_0. By Lemma <ref> and the γ-Fubini embedding in Proposition <ref>, T_K is also bounded as an operator
 
    T_K L^p(^d;γ(^d;X)) →  L^p,∞(^d;γ(^d×^d;Y))

 with norm C_p τ_2,Y A_0. Combined with (<ref>) this implies that ∑_jT_KS_jb_j is well-defined.



Using these auxiliary operators we now decompose as follows:

    T_K b_γ(^d;Y)>λ/2   = T_ψb_γ(^d×^d;Y)>λ/2
       ≤T_ψ b - ∑_jT_KS_jb_j_γ(^d×^d;Y)>λ/4∖ O
            +∑_jT_KS_jb_j_γ(^d×^d;Y)>λ/4 +O
       =: [15pt]A+[15pt]B+[15pt]C

To estimate [15pt]A we first note that by Chebyshev's inequality and Lemma <ref> we have

    [15pt]A≤τ_2,Y^2 16/λ^2∫_^d ∖ OT_ψ b-∑_jT_KS_jb_j_L^2(^d×^d;Y)^2

Using the fact that the b_j's are disjointly supported on the cubes Q_j ⊆ B_j, Fubini's theorem, the 2-Hörmander condition and (<ref>) we deduce

    ∫_^d ∖ OT_ψ b-∑_jT_KS_jb_j_L^2(^d ×^d;Y)^2
       ≤∑_j   ∫_^d ∖ B_j'∫_B_j∫_^dK(s,t) - K(s,t')ψ_r_j(t',t)b_j(t)_Y^2 t'    t  s
      
       ≤∑_j∫_^d∫_B_j∫_^d ∖ B_j'K(s,t) - K(s,t')^2b_j(t)_X^2  s  t·sup_t ∈ B_jψ_r_j(t',t)^2  t'
       ≤ C_d  K_2^2 ∑_jb_j_L^2(^d;X)^2.

Therefore by the norm estimate of the b_j's in terms of f we have

    [15pt]A   ≤ C_d τ_2,Y^2  K_2^2 f_L^2(^d;X)^2/λ^2.
    For [15pt]B we use the boundedness of T_K and  (<ref>) to obtain[15pt]B   ≤ C_p (τ_2,Y A_0)^p/λ^p∑_j S_jb_j_L^p(^d;γ(^d;X)))^p 
       ≤ C_p,dτ_2,Xτ_2,Y A_0^p μ^p-2f_L^2(^d;X)^2/λ^2and for [15pt]C we have by the estimate of Q_j in terms of f that[15pt]C   ≤∑_jB_j'≤ C_d ∑_jQ_j≤ C_d μ^-2f_L^2(^d;X)^2/λ^2.


Plugging the estimate for g and the estimates for b into (<ref>), we now have for all compactly supported f ∈ L^2(^d;X)∩ L^p(^d;X) that

    T_Kf_L^2,∞(^d;γ(^d;Y))≤ C_p,dτ_2,Xτ_2,YA_0 μ^p+1/μ + τ_2,YK_2f_L^2(^d;X),

where we used that τ_2,X,τ_2,Y≥ 1. By density this estimate extends to all f ∈ L^2(^d;X), so choosing μ := (τ_2,Xτ_2,YA_0)^-1 finishes the proof of the weak L^2-endpoint.



In general one cannot expect T_K ∈K_γ(L^2(^d)) in Theorem <ref> , which is already clear from the scalar case. For instance the kernel K(s,t) = 1/(s+t)^1/2_s,t>0 of Example <ref> is a  2-Hörmander kernel. However, L^p-boundedness holds only for p∈ (2, ∞).




 §.§ Extrapolation for p<q<infty

We now turn our attention to extrapolation upwards for γ-integral operator. We will show that if K ∈K_γ(L^p,∞(^d)) satisfies the 2-Hörmander condition, then also K ∈K_γ(L^q(^d)) for all q ∈ (p,∞) and we will prove a -endpoint result.



Let X and Y be Banach spaces and assume that Y has type 2. Let p ∈ [2,∞) and suppose K ∈K_γ(L^p,∞(^d)) satisfies the 2-Hörmander condition. Then

 
  *  K ∈K_γ(L^q(^d)) for all q ∈ (p,∞) with
 
    K_K_γ(L^q(^d))   ≤ C_p,q,d K_K_γ(L^p,∞(^d)) + τ_2,YK_2.


  *  There exists a T_K ∈L(L^∞(^d;X), (^d;γ(^d;Y))) such that
    
    T_K_L^∞→   ≤ C_p,d K_K_γ(L^p,∞(^d))  + τ_2,YK_2.

  and T_Kf - T_Kf is constant for all f ∈ L^p(^d;X)∩ L^∞(^d;X).
 


The extension of T_K to all f ∈ L^∞(^d;X) in Theorem <ref><ref> is not in the traditional sense, as even for f ∈ L^p(^d;X)∩ L^∞(^d;X) the extension T_Kf may not coincide with T_Kf. However, as T_Kf and T_Kf only differ by a constant in this case, they represent the same function in the Banach space 
    (^d;γ(^d;Y))/γ(^d;Y).
 Furthermore we cannot claim uniqueness, as L^p(^d;X)∩ L^∞(^d;X) is not dense in L^∞(^d;X)


In order to prove Theorem <ref>, we need to introduce local versions of the operator T_K. For any cube Q ⊆^d we define the local operator

    T^Q_K:L^∞(^d;X) → L^p(Q;γ(^d;Y))

for  s ∈ Q and φ∈ L^2(^d) by

    T^Q_Kf(s)φ:= T_K(_Bf)(s)φ + _Q ∫_^d ∖ BK(s,t)-K(s',t)f(t)φ(t) t s',

where B is the ball with the same center as Q and twice the diameter of Q. Note that T^Q_K is well-defined since _Bf ∈ L^p(^d;X) and for a.e. s,s' ∈ Q we have
  
    K(s,·)- K(s',·)_^d ∖ Bf_γ(^d;Y)≤τ_2,YK_2f_L^∞(^d;X).

  by Lemma <ref>. Heuristically one may think about T^Q_K as
  
    T^Q_Kf(s)= T_K(f)(s)+ _Q K(s',·)f(·)_^d ∖ B(·) s',

 which is of course not well-defined in general.

   These operators satisfy the following properties:


Let X and Y and be Banach spaces and assume that Y has type 2. Let p ∈ [2,∞) and  suppose K ∈K_γ(L^p,∞(^d)) satisfies the 2-Hörmander condition. Let Q, Q'⊆^d be cubes, then

  
  *  for all  f ∈ L^∞(^d;X) we have

    T^Q_Kf_L^p,∞(Q;γ(^d;Y))≤ C_p,dK_K_γ(L^p,∞(^d)) + τ_2,YK_2Q^1/pf_L^∞(^d;X).

  
  *  for all  f ∈ L^p(^d;X) ∩ L^∞(^d;X) there exists a c ∈γ(^d;Y) such that

    T_Kf(s) -T^Q_Kf(s)    = c

for a.e.  s ∈ Q.
 
  *  for all  f ∈ L^∞(^d;X) there exists a c ∈γ(^d;Y) such that

    T^Q_Kf(s) - T_K^Q'f(s)    = c

  for a.e. s ∈ Q∩ Q',




   Let B,B'⊆^d be the balls with the same center as Q,Q' and twice the diameter of Q,Q'. Take f ∈ L^∞(^d;X), then by the assumption on T_K we have
  
    T_K(_Bf)_L^p,∞(^d;γ(^d;Y))   ≤K_K_γ(L^p,∞(^d))_Bf_L^p(^d;X)
       ≤ C_d K_K_γ(L^p,∞(^d))Q^1/pf_L^∞(^d;X).

  Since _Q_L^p,∞(^d) = C_p Q^1/p, <ref> now readily follows using the definition of T^Q_Kf and (<ref>).

  Next take f ∈ L^p(^d;X) ∩ L^∞(^d;X) and let s,s' ∈ Q. Then if we define c := _Q T_K(_^d ∖ Bf)(s') s', we have for a.e. s ∈ Q that

    T_Kf(s)    = T_K(_B f)(s)  +  T_K(_^d ∖ Bf)(s) - _Q T_K(_^d ∖ Bf)(s') s' + c
       = T_K^Qf(s) +c

proving <ref>.

For <ref> by considering a cube Q” containing both Q and Q' we may assume without loss of generality that Q'⊆ Q and thus also B'⊆ B. Fix φ∈ L^2(^d) and define

    g(s,s',t):= K(s,t)-K(s',t) f(t) φ(t).

Then we have for a.e. s ∈ Q∩ Q' by Fubini's theorem
  
    T^Q_K   f(s)φ - T_K^Q'f(s)φ
       = T_K(_B∖ B'f)(s)φ
        + _Q ∫_^d ∖ B - _Q'∫_B ∖ B' - _Q'∫_^d ∖ B g(s,s',t) t s'
       = _Q' T_K(_B∖ B'f)(s')φ s'+ ∫_^d ∖ B_Q  -_Q' g(s,s',t) s' t
       = _Q' T_K(_B∖ B'f)(s')φ s'- ∫_^d ∖ B_Q  -_Q' K(s',t) f(t) φ(t)  s' t.

As the final right-hand side does not depend on s, this proves <ref>.


Using the properties of these local operators T_K^Q we can prove an L^∞-estimate of T_K involving the sharp maximal operator.

  
Let X and Y be Banach spaces and assume that Y has type 2. Let p ∈ [2,∞) and suppose K ∈K_γ(L^p,∞(^d)) satisfies the 2-Hörmander condition. Then for all f  ∈ L^p(^d;X) ∩ L^∞(^d;X) we have

    M^#T_Kf_L^∞(^d)≤ C_d  K_K_γ(L^p,∞(^d)) + τ_2,YK_2f_L^∞(^d;X).




  Let Q⊆^d be a cube and let f ∈ L^p(^d;X) ∩ L^∞(^d;X). Using Lemma <ref><ref>, choose c ∈γ(^d;Y) such that for a.e. s∈ Q
  
    T_Kf(s)-T^Q_Kf(s)    = c.

  Therefore using (<ref>) and Lemma <ref><ref>, we have
  
    _QT_Kf(s)-_Q    T_Kf(t) t_γ(^d;Y) s 
       ≤ 2_QT_Kf-c_γ(^d;Y)
       = 2_QT^Q_Kf_γ(^d;Y)
       ≤ 2Q^-1+1/p' T^Q_Kf_L^p,∞(Q;γ(^d;Y))
       ≤ C_d K_K_γ(L^p,∞(^d)) + τ_2,YK_2f_L^∞(^d;X).

   Therefore we have
  
    M^#T_Kf_L^∞(^d)≤ C_d (K_K_γ(L^p,∞(^d)) + τ_2,YK_2 ) f_L^∞(^d;X),

  which proves the proposition.


Using Proposition <ref>, the proof of Theorem <ref><ref> is now a straightforward application of Stampacchia interpolation (see e.g. <cit.>).


Let f ∈ L^p(^d;X) ∩ L^∞(^d;X). Note that trivially M^#f ≤ 2 M (f_γ(^d;Y)), so by Lemma <ref> we know that M^# is bounded from  L^p,∞(^d;γ(^d;Y)) to L^p,∞(^d). Thus

    M^#T_Kf_L^p,∞(^d)   ≤ C_p,dT_Kf_L^p,∞(^d;γ(^d;Y))
       ≤ C_p,d K_K_γ(L^p,∞(^d))f_L^p(^d;X).
    Moreover by Proposition <ref> we know thatM^#T_Kf_L^∞(^d)   ≤ C_p,d K_K_γ(L^p,∞(^d))  + τ_2,YK_2f_L^∞(^d;X),

We can therefore apply the Marcinkiewicz interpolation theorem (see e.g. <cit.>), to conclude that for all f ∈ L^p(^d;X)∩ L^q(^d;X) we have

    M^#T_Kf_L^q(^d)≤ C_p,q,dK_K_γ(L^p,∞(^d)) +τ_2,YK_2f_L^q(^d;X).

By Lemma <ref> we deduce

    T_Kf_L^q(^d;γ(^d;Y))   ≤ C_q,d M^#T_Kf_L^q(^d)
       ≤ C_p,q,dK_K_γ(L^p,∞(^d)) +τ_2,YK_2f_L^q(^d;X).

for all f ∈ L^p(^d;X)∩ L^q(^d;X). As this is a dense subspace of L^q(^d;X), assertion <ref> of Theorem <ref> follows.


Assertion <ref> of Theorem <ref> does not follow directly from Proposition <ref>, since L^p(^d;X) ∩ L^∞(^d;X) is not dense in L^∞(^d;X) and therefore the extension of T_K to all functions in  L^∞(^d;X) is a nontrivial matter.


 Let (Q_k)_k=1^∞ be an increasing sequence of cubes such that ⋃_k=1^∞ Q_k = ^d. For f ∈ L^∞(^d;X) define
  
    T_Kf(s):=T^Q_k_Kf(s)- _Q_1T^Q_k_Kf    if  s ∈ Q_k.

  Then T_Kf ∈ L^1_(^d;γ(^d;Y)) is well-defined. Indeed, by Lemma <ref><ref> we have T^Q_k_Kf ∈ L^1(Q_k;γ(^d;Y)), so in particular the average over Q_1 is well-defined. Moreover if j>k, then by Lemma <ref><ref>  there is a c ∈γ(^d;Y) such that T^Q_j_Kf(s) -T^Q_k_Kf(s)= c for a.e. s ∈ Q_k. Therefore
  
    T^Q_k_Kf(s)- _Q_1T^Q_k_Kf    = T^Q_j_Kf(s)-c - _Q_1 T^Q_j_Kf - c
       = T^Q_j_Kf(s) - _Q_1 T^Q_j_Kf,

  thus the definition of T_Kf(s) is independent of the choice of Q_k ∋ s.

  If f ∈ L^p(^d;X)∩ L^∞(^d;X), then for any k ∈ there exist c_1,c_2 ∈γ(^d;Y) such that for a.e. s ∈ Q_k
  
    T^Q_k_Kf(s)-T_Kf(s)    = c_1,
    
        T^Q_k_Kf(s)-T_Kf(s)    = c_2

  by Lemma <ref><ref> and the definition of T_Kf. As (Q_k)_k=1^∞ is increasing and ⋃_k=1^∞ Q_k = ^d, we see that c_1 and c_2 are independent of k, so T_Kf-T_Kf is indeed constant.

  It remains to show that T_Kf ∈(^d;X) with the claimed norm estimate. Let Q⊆^d be any cube and fix k ∈ such that Q ⊆ Q_k. By Lemma <ref><ref> there exists a c_3 ∈γ(^d;Y) such that for a.e. s∈ Q
  
    T^Q_k_Kf(x)-T^Q_Kf(s)    = c_3.

  Therefore
  
    T_Kf_(^d;γ(^d;Y))≤_QT_Kf-(c_3-c_2)_γ(^d;Y) = _QT^Q_Kf_γ(^d;Y).

  Now _QT^Q_Kf_γ(^d;Y) can be estimated exactly as in the proof of Proposition <ref>, which yields the claimed norm estimate in Theorem <ref><ref>.



  By inspection of the proof it can easily be seen that for the extrapolation down in Theorem <ref> one only needs
  
    ∫_^d ∖ BK(s,t)-K(s,t')x_Y^2 t^1/2≤ C  x_X,      s,s'∈12B ,    x ∈ X

  which is implied by (<ref>) of the 2-Hörmander condition. For the extrapolation up in Theorem <ref> one only needs the left hand side of (<ref>) to be bounded, which is implied by (<ref>) of the 2-Hörmander condition.




  * In <cit.> a real-valued extrapolation result was proved under a parabolic Hörmander condition which allows one to extend L^2(Ω×_+× D)-boundedness to L^p(Ω×_+× D)-boundedness. In applications to SPDEs this result can be combined with ours to extrapolate L^2(Ω×_+× D)-boundedness to L^p(Ω×_+;L^q(D))-boundedness for all p∈ (2, ∞) and q∈ [2, ∞) (see Examples <ref> and <ref>).

  * Another type of BMO end-point estimate was obtained in <cit.>, but the result seems incomparable with ours.




Let X be a Banach space and Y be a Hilbert space. Suppose k:^d→L(X,Y) is strongly measurable and satisfies the 2-Hörmander condition in (<ref>).
Let K(s,t) = k(s-t). Then the following are equivalent:


  *  t↦ k(t)x_L^2(^d;Y)≤ A_0  x for some A_0>0;

  *  K ∈K_γ(L^p(^d)) for all p∈ [2, ∞).

  *  K ∈K_γ(L^p,∞(^d)) for some p∈ [2, ∞);

In particular we have for all p ∈ [2,∞) and A_0 as in <ref>:

    K_K_γ(L^p(^d))≤ C_p,d  (A_0 + K_2).



The implication <ref> ⇒ <ref> for p=2 follows from Proposition <ref><ref> and for p∈ (2, ∞) we can apply Theorem <ref>. The implication <ref> ⇒ <ref>  is trivial and <ref> ⇒ <ref> follows from Proposition <ref>.




§ SPARSE DOMINATION FOR Y-INTEGRAL OPERATORS

In this section we will obtain weighted bounds for a γ-integral operator T_K under an (ω,2)-Dini condition on K. We will deduce these weighted bounds by estimating the operator T_K pointwise by a much simpler operator. These simpler operators satisfy weighted bounds, which then imply weighted bounds for T_K.

Let us start by defining these simpler operators. We say that a collection of cubes S in ^d is η-sparse for some η∈ (0,1) if for every Q ∈S there exists an E_Q ⊆ Q such that E_S≥ηS and such that the collection E_Q:Q ∈S is pairwise disjoint. Typically η will only depend on the dimension d. We will dominate the γ-integral operators by operators of the form

    f ↦∑_Q ∈S_Q f_X^2_Q^1/2

for some η-sparse collection of cubes S. These sparse operators are well-known to satisfy weighted bounds, see <cit.>.

The sparse domination approach was developed in order to prove the so called A_2-conjecture, first solved by Hytönen in <cit.>. The particular result we need stems from Lacey's simple proof of the A_2-conjecture <cit.>, further clarified and simplified by Lerner <cit.> and later by Lerner and Ombrosi <cit.>. We will use a version of this result by the first author <cit.>, which is adapted to our stochastic vector-valued setting.
In order to use this result we need to study a grand maximal truncation operator associated to  the operator that we wish to dominate by a sparse operator.

Let X and Y be Banach spaces, p ∈ [1,∞) and let T be a bounded operator from L^p(^d;X) to L^p,∞(^d;Y). Define for f ∈ L^p(^d;X), α≥ 1 and s ∈^d

    M_T,α^#f(s) := sup_B ∋ s _s',s”∈ B T(_^d ∖α Bf)(s')-T(_^d ∖α Bf)(s”)_Y,

where the first supremum is taken over all balls B ⊆^d containing s and α B is the dilation of B by a factor α. The following theorem is a special case of <cit.>.


Let X and Y be a Banach spaces, p,r∈ [1,∞) and α≥ 6. Assume the following conditions:

  
  * T is a bounded linear operator from L^p(^d;X) to L^p,∞(^d;Y).
  
  * M_T,α^# is bounded from L^p(^d;X) to L^p,∞(^d).
  
  * For disjointly supported f_1,⋯,f_n ∈ L^p(^d;X) we have

    T∑_k=1^n f_k(s)_Y≤ C_r  ∑_k=1^n Tf_k(s)_Y^r^1/r,
           s ∈^d.


 Then there exists an η∈ (0,1) such that for any compactly supported f ∈ L^p(^d;X) there is an η-sparse collection of cubes S such that

    Tf(s)_Y   ≲_α,d C_T  C_r ∑_Q ∈S_Qf_X^p^r/p_Q(s)^1/r,      s ∈^d,

where C_T=T_L^p_1→ L^p_1,∞ + M_T,α^#_L^p_2→ L^p_2,∞.


In order to apply this theorem on a K ∈K_γ(L^p(^d)) we need to check weak L^2-boundedness of T_K and M_T_K,α^# and we need to check  equation (<ref>) with r=2.
For a (ω,2)-Dini kernel the boundedness of M_T_K,α^# is quite easy to check for α=6:

   Let X and Y be Banach spaces and assume that Y has type 2. Let p ∈ [2,∞)  and suppose K ∈K_γ(L^p,∞(^d)) satisfies the (ω,2)-Dini condition. Then for any f ∈ L^p(^d;X) we have
   
    M^#_T_K,6f  ≤ C_d τ_2,Y K_ω2 M_2(f_X).

 In particular M^#_T_K,6 is bounded from L^2(^d;X) to L^2,∞(^d)  with

    M^#_T_K,6_L^2(^d;X) → L^2,∞(^d)   ≤ C_d τ_2,Y K_ω2.




Let f ∈ L^p(^d;X), s ∈^d and fix a ball B ∋ s with radius ρ. Take s',s”∈ B and let ε = 4ρ. Then s'-s”≤1/2ε and for any t ∈^d∖ 6 B we have s'-t > ε. Therefore applying Lemma <ref> and using the (ω,2)-Dini condition we obtain
  
    T_K(_^d ∖ 6B   f)(s') - T_K(_^d ∖ 6Bf)(s”)_γ(^d;Y)
       ≤τ_2,Y∫_^d∖ 6BK(s',t) - K(s”,t)f(t)^2_Y t^1/2
       ≤τ_2,Y∫_s'-t>εωε/2/s'-t^21/s'-t^df(t)_X^2 t^1/2
       ≤τ_2,Y∑_j=0^∞ω(2^-j-1)^2/(2^jε)^d∫_2^jε<s'-t≤ε2^j+1f(t)_X^2 t^1/2
       ≤ C_d  τ_2,Y∑_j=0^∞ω(2^-j-1)^2 _B(s',2^j+1ε)f(t)_X^2 t^1/2
       ≤ C_d  τ_2,Y K_ω2   M_2f_X(s) .

    where the last step follows from s ∈ B(s',2^j+1ε) for all j ∈.
  Now taking the essential supremum over s',s”∈ B and the supremum over all balls B ∋ s, we see that
  
    M^#_T_K,6f(s) ≤  C_d  τ_2,Y K_ω2   M_2f_X(s),      s ∈ S.

  The weak L^2-boundedness follows from the corresponding bound for M_2 in Lemma <ref> and the density of L^p(^d;X) in L^2(^d;X).


We can now prove sparse domination, and thus also weighted boundedness, for the γ-integral operators


Let X and Y be Banach spaces with type 2. Let p ∈ [2,∞) and suppose K ∈K_γ(L^p,∞(^d)) satisfies the (ω,2)-Dini condition. Then there is an η∈ (0,1) such that for every compactly supported f ∈ L^2(^d;X) there exists an η-sparse collection of cubes S such that

    Tf(s)_γ(^d;Y)≤ C_X,Y,p,d c_K ∑_Q ∈S_Q f_X^2_Q(s)^1/2,      s ∈^d

with c_K:= K_K_γ(L^p,∞(^d)) + K_ω2.
In particular:

  
  * K ∈K_γ(L^q(^d,w))
for all q ∈ (2,∞) and w ∈ A_q/2 with
  
    K_K_γ(L^q(^d,w))   ≤ C_X,Y,p,q,d c_K  [w]_A_q/2^max1/2,1/q-2.
      
      * K ∈K_γ(L^2,∞(^d,w))
    for all w ∈ A_1 withK_K_γ(L^2,∞(^d,w))   ≤ C_X,Y,p,d c_K  [w]_A_1^1/2log1+[w]_A_∞^1/2.





Since K is an (ω,2)-Dini kernel, it is also a 2-Hörmander kernel by Lemma <ref> with

    K_2≤ C_p,d K_ω2.

Therefore by Theorem <ref> we know that T is bounded from L^2(^d;X) to L^2,∞(^d;γ(^d;Y)) with norm

    T_L^2 → L^2,∞≤ C_p,d τ_2,Xτ_2,YK_K_γ(L^p,∞(^d)) + τ_2,YK_ω2.

By Lemma <ref> we  also know that M^#_T,6 is bounded from L^2(^d;X) to L^2,∞(^d) with norm

    M_T,6^#_L^2 → L^2,∞≤ C_d τ_2,YK_ω2.

Moreover for f_1,⋯,f_n ∈ L^2(^d;X) with disjoint support we have for a.e. s ∈^d that T_Kf_1(s)⋯ T_Kf_n(s) have disjoint support as well and thus (<ref>) with r=2 follows from Lemma <ref>. The sparse domination therefore follows by applying Theorem <ref> to T_K.
The weighted bounds follow directly from <cit.> and the density of compactly supported L^2-functions in L^q(^d,w;X) for all q ∈ [2,∞).


 

  
  *  If we omit the type 2 assumption for X in Theorem <ref> we can still conclude that T_K is sparsely dominated by larger sparse operator
      
    f ↦∑_Q ∈S_Q f_X^p^p/2_Q^1/2

 In the proof one then has to skip the step where Theorem <ref> is applied. This is in particular useful when p=2.
  
  * A_p/2 is the largest class of weights one can expect in Theorem <ref>, since in the case that X=Y= and K(s,t)=k(s-t), Theorem <ref> can be reduced to a statement about deterministic convolution operators with positive kernel (see Subsection <ref>).
      It is standard to check that the weighted boundedness of for example

    Tf(s) :=  ∫_^dλ^d ^-λ|s-t| f(t)  t,      s ∈^d,

for all λ∈_+ implies the A_p-condition, see e.g. <cit.>. Also the dependence on the weight characteristic is sharp, see Proposition <ref> below



Under a Dini type condition we obtain the following further characterization if Y is a Hilbert space. The proof is immediate from Corollary <ref>, Theorem <ref> and Remark <ref><ref>.

Let X be a Banach space and Y be a Hilbert space. Suppose k:^d→L(X,Y) is strongly measurable and satisfies the (ω,2)-Dini condition in (<ref>).
Let K(s,t) := k(s-t). Then statements <ref>–<ref> in Corollary <ref> are equivalent to


  *  K ∈K_γ(L^p(^d,w)) for all p∈ (2, ∞) and all w∈ A_p/2.

In particular we have for all p ∈ (2,∞), w ∈ A_p/2 and A_0 as in <ref> of Corollary <ref>:

    K_K_γ(L^p(^d))≤ C_p,d  (A_0 + K_ω2) [w]_A_q/2^max1/2,1/q-2.



We will show next that the dependence on the weight characteristic [w]_A_p/2 in the bounds for T_K in Theorem <ref> is actually optimal. Therefore Theorem <ref> can be thought of as a γ-analog of the A_2-theorem in the deterministic setting.
 
Let X and Y be Banach spaces and p∈ (2,∞) and β≥ 0. There exists a kernel K^d →L(X,Y) satisfying the assumptions of Theorem <ref> such that if for all w ∈ A_p/2 we have

    K_K_γ(L^p(^d,w))≲ [w]_A_p/2^β,

then β≥max12,1q-2.



  By considering one dimensional subspaces, we may assume without loss of generality that X=Y=. Define

    K(s_1,s̅),(t_1,t̅) :=
      (s_1+t_1)^1/2/(s_1,s̅)+(t_1,t̅)^(d+1)/2,      (s_1,s̅),(t_1,t̅)∈×^d-1.

Then by Lemma <ref> we know that K is a (1,2)-standard kernel.

Set _+^d:= (s_1,s̅) ∈×^d-1: s_1≥ 0, _-^d:= ^d ∖^d_+ and define for q ∈ (1,∞) and f ∈ L^q(^d_+)

    T_1f(s) := Γ((d+1)/2)/π^(d+1)/2∫__+^ds_1+t_1/s+t^d+1f(t)  t,      s ∈^d_+.

Then T_1 is a bounded operator on L^q(_+^d) for all q ∈ (1,∞) with

    T_1_L^q(^d_+)→ L^q(^d_+) = 1/sinπ/q,

by <cit.>.
For g ∈ L^p(^d) we have

    T_K g_L^p(^d)^p    = T_K (g_^d_++g _^d_-)_L^p(^d_+)^p +T_K (g_^d_++g _^d_-)_L^p(^d_-)^p
       _d  T_1 h _L^p/2(^d_+)^p/2

where h(s):= |g(s)+g(-s)|^2 for s ∈^d_+. Therefore

    K_K_γ(L^p(^d))_d T_1_L^p/2(^d_+)→ L^p/2(^d_+)^1/2 = 1/sin2π/p^1/2,

so K satisfies the assumptions of Theorem <ref>. Moreover
  
    α_K   := supα≥ 0: ∀ε>0, lim sup_p→ 2K_K_γ(L^p(^d))/(p-2)^-α+ε=∞ = 1/2
    γ_K   := supγ≥ 0: ∀ε>0, lim sup_p→∞K_K_γ(L^p(^d))/p^-γ+ε=∞ = 1/2.

Thus by <cit.> it follows that if

    K_K_γ(L^p(^d,w))≲ [w]_A_p/2^β,

then
  
    β≥maxα_K  2q-2, γ_K= max1q-2,12.








§ EXTENSION TO SPACES OF HOMOGENEOUS TYPE

In this section we will describe how Sections <ref>-<ref> can be generalized from ^d to a space of homogeneous type (S,d,μ). This will be quite useful in our applications, as we will often want to take S = _+ or S = (0,T) with the Euclidean metric and the Lebesgue measure. While these examples are in a sense trivial spaces of homogeneous type, they do not follow directly from our theory on ^d.

A space of homogeneous type (S,d,μ), originally introduced by Coifman and Weiss in <cit.>, is a set S equipped with a quasimetric d, i.e. a metric which satisfies

    d(s,t) ≤ c_d d(s,u)+d(u,t),      s,t,u∈ S

for some c_d ≥ 1 instead of the triangle inequality, and a Borel measure μ that satisfies the doubling property, i.e.

    μB(s,2r)≤ c_μ μB(s,r),      s ∈ S,   r>0

for some c_μ≥ 1. In addition we assume that all balls B⊆ S are Borel sets and that we have 0 <μ(B)<∞. As μ is a Borel measure the Lebesgue differentiation theorem holds and C_c(S) is dense in L^p(S) for all p ∈ [1,∞), see <cit.> for the details.



We will now describe how Sections <ref>-<ref> can be adapted to spaces of homogeneous type. When we say that a result remains valid when we replace ^d with a space of homogeneous type (S,d,μ), we mean implicitly that all cubes are replaced by balls with the same center and diameter and that the dependence on the dimension d of the involved constants is replaced by dependence on the quasimetric constant c_d and the doubling constant c_μ.


  * The weighted bounds for the Hardy–Littlewood maximal operator in Lemma <ref> are still valid, since we can do a covering lemma argument similar to the one we did for ^d, see <cit.>.

  * The definition of the 2-Hörmander condition carries over directly to spaces of homogeneous type. For the (ω,2)-Dini we replace s-t^d by μ(B(s,d(s,t))) in (<ref>) and by μ(B(t,d(t,s))) in (<ref>).

  * Lemma <ref> remains valid in general spaces of homogeneous type and Lemma <ref> as well if S is a convex subset of ^d with the Euclidean distance and the Lebesgue measure. More generally, Lemma <ref> also remains valid if S is a smooth domain in ^d, as one can then locally reduce to the ^d_+ case. Lemma <ref> remains true for (0,T) with T∈ (0,∞].
  
  * Theorem <ref> remains valid. The main part of the proof that should be adapted, is the L^2-Calderón–Zygmund decomposition.
       This decomposition in spaces of homogeneous type can be found in <cit.> in the case X = and the proof again carries over verbatim to the vector-valued setting. Note that this decomposition at level λ>0 holds only when
      
    λ≥_S f^2 μ^1/2,

  where the right hand side is of course zero if μ(S) = ∞. So if μ(S) <∞ we need another argument in the proof of Theorem <ref> in the case
  
    μλ≤_S f_X^2 μ^1/2.

  But this case is trivial, since
  
    T_Kf_γ(^d;Y)>λ^1/2   ≤μ(S)^1/2≤1/μλf_L^2(S;X).

  The other difference in this decomposition is that we do not obtain a disjoint decomposition of the “bad” part b, but a decomposition with bounded overlap.  One can easily check that this does not cause any problems in our proof. For instance in the inequality
  
    ∑_j _Q_j_L^p(^d)≤∑_jQ_j^1/p

  one needs to add a constant depending on the amount of overlap.
  
  * Theorem <ref> also remains valid. The main difficulty here is the Fefferman-Stein theorem (Lemma <ref>) in spaces of homogeneous type, which can be found in  <cit.> or <cit.>. When μ(S) < ∞, this requires some extra care, since we then have
      
    f_L^q(^d)≲_q,c_d,c_μ M^#f_L^q(^d) + μ(S)^-1/q'f_L^1(^d).

      In the proof of Theorem <ref> this means that we also need to estimate μ(S)^-1/q'T_Kf_L^1(S) in terms of f_L^q(^d;X). By the assumption that K ∈K_γ(L^p,∞(S)), Hölder's inequality and (<ref>) we have
      
    μ(S)^-1/q'T_Kf_L^1(S;Y)   ≤ C_p  μ(S)^1/p'-1/q'T_Kf_L^p,∞(S;Y)
       ≤ C_p  μ(S)^1/p'-1/q'K_K_γ(L^p,∞(S;X))f_L^p(S;X)
       ≤ C_p  K_K_γ(L^p,∞(S))f_L^q(S;X),

      which is exactly the required estimate.
  
  * The proof of Theorem <ref> relies completely on the results in <cit.>, which are proven in a space of homogeneous type. Therefore Theorem  <ref> remains valid.
    
  * Corollary <ref> and Corollary <ref> remain valid on _+.





§ APPLICATIONS TO STOCHASTIC MAXIMAL REGULARITY

We will now apply our results from Sections <ref>-<ref> to obtain stochastic maximal regularity of various SPDE's. For this we will first need some background on sectorial operators.



 §.§ Sectorial operators

Let X be a Banach space and define

    Σ_φ = {z∈∖{0}: |(z)|<φ}.

A closed operator (A,D(A)) on X will be called sectorial if there is a φ∈ [0,π) such that ∖Σ_φ⊆ρ(A) and there is a constant A_0>0 such that

    (λ-A)^-1≤ A_0/|λ|,   λ∈∖Σ_φ.

The infimum over all such φ is called the angle of sectoriality of A.
For details on sectorial operators we refer to <cit.>. In particular, recall that for a sectorial operator one can define the fractional powers A^α for α∈.
For θ>0 and p∈ [1, ∞], the spaces D_A(θ,p) are defined by

    D_A(θ,p) = (X, D(A^n))_θ/n,p,

where n∈ is the least integer larger than θ. In the above we used the real interpolation method.  The complex interpolation method will be used as well, and our notation for this will be [X, D(A)]_θ. For details on interpolation we refer to <cit.>.




 §.§ Setting

Many stochastic PDEs can be analyzed as stochastic evolution equations by using functional analytic tools. We refer to the monograph <cit.> and the papers <cit.>.

Consider the following linear stochastic evolution equation on a Banach space X:

    du + A u t = G  W_H      on _+,
    
    u(0)=0.

Here (A(t))_t∈_+ is a family of closed operators on X, W_H is H-cylindrical Brownian motion and G:_+×Ω→γ(H,X) is adapted to F. In this paper we will focus on linear equations. Nonlinear stochastic evolution equations can be studied by using suitable estimates for the linear case (see <cit.>). In particular, stochastic maximal regularity estimates have been applied to nonlinear SPDEs in <cit.>.

The mild solution to (<ref>) is given by

    u(t) = S♢ G(t):=∫_0^t S(t,s) G(s) W_H(s),      t ∈_+.

Here we have assumed that -A generates the strongly continuous evolution family (S(t,s))_0≤ s≤ t. In the case A does not depend on time, one has that S(t,s) = e^-(t-s)A is a strongly continuous semigroup. For details and unexplained terminology on semigroups and evolution families we refer to <cit.>.


Let X be a  Banach space with type 2, p∈ [2, ∞) and let w be a weight on _+. Let Y↪ X. We say that A has stochastic maximal L^p(w;Y)-regularity and write A ∈(L^p(w;Y)) if for all G∈ L^p_F(Ω×_+,w;γ(H,X)) the mild solution u to (<ref>) satisfies

    u_L^p(Ω×_+,w;Y)≤  C G_L^p(Ω×_+,w;γ(H,X)).
We omit the weight if w ≡ 1.

Written out explicitly the estimate (<ref>) becomes

    t↦∫_0^t S(t,s) G(s) W_H(s)_L^p(Ω×_+,w;Y)≤ C  G_L^p(Ω×_+,w;γ(H,X)).

Interesting choices for Y are the complex and real interpolation spaces

    Y  = [X, D(A)]_1/2  and   Y = (X, D(A))_1/2,r,   r∈ [2, ∞),

and the fractional domain spaces Y = D((A+λ)^1/2) for λ such that A+λ is sectorial. In several places we will use the homogenous fractional domain space Ḋ((A+λ)^1/2) with norm

    x_Ḋ((A+λ)^1/2) = (A+λ)^1/2 x_X.

Note that if A+λ is invertible, then Ḋ((A+λ)^1/2) = D((A+λ)^1/2).

In <cit.> it has been shown that under certain geometric restrictions on X, the boundedness of the H^∞-calculus of angle <π/2 of A (see <cit.>) implies

    A∈(L^p(Ḋ(A^1/2))).

Extensions to the case of time-dependent A have been obtained in <cit.>. Abstract properties of stochastic maximal regularity have been studied in <cit.>, where in particular it was shown that if A is time-independent and A∈(L^p(D(A^1/2))), then -A generates an exponentially stable analytic semigroup.
In case A  has a bounded H^∞-calculus of some angle, then one has [X, D(A)]_1/2 = D(A^1/2) (see <cit.>).

Stochastic maximal regularity can equivalently be formulated using the stochastic integral operators of Definition <ref>. In this case the kernel is given by

    K(t,s) :=  S(t,s) _0≤ s < t∈L(X,Y).

Here we implicitly assume that S(t,s) maps X into Y.
Below we will apply the extrapolation theory of Section <ref> to study independence of p and the weight for Definition <ref>.



 §.§ Semigroup case
 We first turn to the time independent case.


Suppose X is a  Banach space with type 2.
Let A be sectorial of angle <π/2. Take r∈ [2, ∞) and assume that Y is one of the spaces

    D(A^1/2),   Ḋ(A^1/2),      [X, D(A)]_1/2,   or   (X, D(A))_1/2,r,

Suppose A∈(L^p(Y)) for some p∈ [2, ∞). Then for all q∈ (2, ∞) and w∈ A_q/2 one has A∈(L^q(w;Y)). In particular, the mild solution u to (<ref>) satisfies

    u_L^q(Ω×_+,w;Y)≤ C  [w]_A_q/2^max{1/2, 1/q-2}G_L^q(Ω×_+,w;γ(H,X)),

where C only depends on X, A, p, q.



The space Y has type 2 with τ_2,Y≤τ_2,X, which is trivial for D(A^1/2) and Ḋ(A^1/2), follows from <cit.> for [X, D(A)]_1/2 and follows from <cit.> for (X, D(A))_1/2,r.

In all cases except for Ḋ(A^1/2) it follows from the proof of <cit.> that A is invertible. We claim that in all cases

    x_Y ≤ C x^1/2_X Ax^1/2_X,       x∈ D(A).

Indeed, this standard interpolation estimate follows from <cit.>, <cit.> and <cit.>. Since tA e^-tA_L(X,Y)≤ M for t≥ 0 and some M>0 (see <cit.>), the above interpolation estimate implies

    A e^-tA_L(X,Y)≤ C M^3/2 t^-3/2,       t≥ 0.

By assumption K∈𝒦_W^H(L^p(_+)).  Applying Propositions <ref> and <ref> we obtain that K∈𝒦_γ(L^p(_+)). Next we will check the conditions of Theorem <ref> for the homogenous space _+ (see Section <ref>).

Let k(t) = ^-tA. By the analyticity of the semigroup and the above estimate, we find that for t>0 and

    k'(t)_L(X,Y) = A e^tA_L(X,Y)≤ C M  t^-3/2.

Therefore, by Lemma <ref> we know that K is a (1, 2)-standard kernel with

    K_(1, 2)- Std≤ C M.

Now Theorem <ref> gives that K∈𝒦_γ(L^q(_+,w)).
Propositions <ref> and <ref> then imply K∈𝒦_W^H(L^q(_+,w)) with the claimed estimate.





  *  Combining Theorem <ref> with <cit.>, similar results as in Theorem <ref> hold on finite time intervals (0,T). Alternatively, this can be deduced by applying Theorem <ref> on (0,T) (see also Section <ref>)

  * In general the result of Theorem <ref> does not hold in the endpoint q=2. A counterexample can be found in <cit.>.

  * Arguing as in the proof of Theorem <ref> but with

    K(t,s) = (t-s)^-θ A^1/2-θ e^-(t-s)A,
 it follows that for any θ∈ (0,1/2) the property A∈_θ(p,∞) introduced in <cit.> is p-independent.

  *  From the proof it is clear that Theorem <ref> holds for any space Y↪ X such that e^-tA:X→ Y with

    e^-tA_L(X,Y)≤ C t^-1/2,       t>0.







Let X be a Hilbert space and let Y be any of the spaces in (<ref>) with r=2. Suppose that A is sectorial of angle <π/2 on X. Then the following are equivalent:


  *  There exists a constant C>0 such that

    t↦ e^-tAx_L^2(_+;Y)≤ C x_X,      x∈ X;


  *  For all p∈ (2, ∞) and all w∈ A_p/2 (and p=2, w≡ 1) we have A∈(L^p(w;Y)).

  *  A∈(L^p(Y)) for some p∈ [2, ∞).




Note that Y is a Hilbert space. For <ref>⇒<ref> define K(t,s)∈L(X,Y) by

    K(t,s) = e^-(t-s)A_s <t.
 From Proposition <ref><ref> we obtain K∈𝒦_γ(L^2(_+)). Therefore K∈𝒦_W^H(L^2(_+)) by Propositions <ref> and <ref>, so the result follows from Theorem <ref>. <ref>⇒<ref> is trivial and <ref>⇒<ref> follows from Proposition <ref> combined with Propositions <ref> and <ref>.


 


  * Corollary <ref><ref> is equivalent to the admissibility of A^1/2 and is connected to the Weiss conjecture, which was solved negatively (See <cit.>, <cit.> and references therein).

  * It is well-known that there exist operators A on a Hilbert space X such that -A generates an analytic semigroup which is exponentially stable and
    
    t↦ A^1/2 e^-tA x_L^2(_+;X)≤ Cx_X,
          but
          cx_X≰t ↦ A^1/2 e^-tA x_L^2(_+;X).

     Such A can be constructed as in <cit.> (see <cit.> for details), and  does not have a bounded H^∞-calculus. On the other hand, Corollary <ref> implies A∈(L^p(w;D(A^1/2))) for all p∈ [2, ∞) and w∈ A_p/2 (with w=1 if p=2), which shows that having a bounded H^∞-calculus is not necessary for stochastic maximal regularity.




Let E be a  Banach space with type 2, let A be sectorial of angle <π/2 and assume 0∈ρ(A). Let θ∈ (0,1) and q∈ [2, ∞). Define X = D_A(θ,q) and
Y = D_A(θ+1/2,q).  Then for all p∈ (2, ∞) and w∈ A_p/2, one has
A∈(L^p(w,Y)) (the case p=2 and w=1 is allowed as well if q=2). In particular, the solution u to (<ref>) satisfies

    A^1/2 u_L^p(Ω×_+,w;D_A(θ,q))≲ [w]_A_p/2^max{1/2, 1/p-2}G_L^p(Ω×_+,w;γ(H,D_A(θ,q))),

where the implicit constant only depends on E, A,θ, p, q.


Note that X is a  Banach space with type 2 by <cit.> and
-A is the generator of an exponentially stable analytic semigroup on X with domain D_A(θ+1,q) by <cit.>. Moreover, we have Y = (X,D_A(θ+1,q))_1/2,q. It follows from <cit.> (see also <cit.>) and <cit.> that A∈(L^q(Y)). Therefore, the required result follows from Theorem <ref>. The claimed norm estimate follows since A^1/2 maps D_A(θ+1/2,q) isomorphically to D_A(θ,q) (see <cit.>).


Next we give a self-contained proof.

First consider the case p=q=2. By Propositions <ref><ref>, <ref> and <ref> and <cit.> it suffices to show

    [15pt]A:=(∫_0^∞A^1/2 e^-t A x_D_A(θ,2)^2 t)^1/2≤ C x_D_A(θ,2).

Since D_A(θ,2) = D_A^2(θ/2,2) (see <cit.>), by <cit.> we can write

    [15pt]A^2    ∫_0^∞A^1/2 e^-t A x_D_A^2(θ/2,2)^2 t
    
       ∫_0^∞∫_0^∞ r^4(1-θ/2)A^5/2 e^-(r+t) A x_E^2  r/r t
    
       ≲∫_0^∞∫_0^∞  (t+r)^-3 r^4(1-θ/2)A e^-r A x^2_E  r/r t
    
         = 2 ∫_0^∞   r^2(1-θ)A e^-r A x_E^2  r/r≃x_D_A(θ,2)^2

which gives the required estimate (<ref>).

From the previous case and Theorem <ref> we obtain stochastic maximal L^p-regularity for p∈ [2, ∞) in the case q=2. Thus using Propositions <ref> and <ref> to take H=, the mapping

    S(G)(t) := ∫_0^t A^1/2e^-(t-s) A G(s) W(s),      t ∈_+

is bounded from L^p(_+;D_A(θ,2)) to L^p(_+×Ω;D_A(θ,2)) for all θ∈ (0,1) and p∈ [2, ∞). By <cit.> one has

    (L^q(_+;D_A(θ-ε,2)),L^q(_+;D_A(θ+ε,2)))_1/2,q = L^q(_+;D_A(θ,q))

for ε∈ (0,min{θ,1-θ}) and the same holds with _+ replaced by _+×Ω. It follows from <cit.> that S is bounded from  L^q(_+;D_A(θ,q)) into L^q(_+×Ω;D_A(θ,q)). Applying
Propositions <ref> and <ref> once more to recover a general cylindrical Brownian motion W_H, we obtain the stochastic maximal regularity for p=q∈ [2, ∞). Now another application of Theorem <ref> gives the result for all required p, q and weights w∈ A_p/2. The claimed norm estimate again follows since A^1/2 maps D_A(θ+1/2,q) isomorphically to D_A(θ,q) (see <cit.>).


 

  
  * By carefully checking the proofs of Theorems <ref> and <ref> (and in particular Proposition <ref>) one sees that Theorem <ref> actually holds for all martingale type 2 spaces E. As mentioned in Remark <ref><ref>, Theorem <ref> holds on finite time intervals as well and in this case we only need that A+λ is sectorial of angle <π/2 for some λ∈.
  
  * Theorem <ref> extends <cit.> and <cit.> to the case where p≠ q and to the weighted setting.  Note that even for w=1 one cannot obtain Theorem <ref> from the case p=q and a real interpolation argument. Indeed, in general for an interpolation couple (X_0, X_1) one has (see <cit.>)

    (L^p_0(Ω×_+;X_0),L^p_1(S;X_1))_θ,q≠ L^p_θ(Ω×_+;(X_0, X_1)_θ,q),

with 1/p_θ = 1-θ/p_0 + θ/p_1. The equality does hold if q=p_θ.


  * The assumption 0∈ρ(A) in Theorem <ref> is needed in general. Indeed, there exists a bounded sectorial operator A on a Hilbert space E such that (<ref>) does not hold (see <cit.>). Since in this case D_A(θ,p) = E for all θ∈ (0,2) and p∈ [1, ∞], Propositions <ref>, <ref> and <ref> imply that (<ref>) cannot hold.



We end this subsection with another result for real interpolation spaces. It extends <cit.> to the case p∈ (2, ∞) and to the setting of infinite time intervals.

Let E be a  Banach space with type 2 and let A be sectorial of angle <π/2 on E. Let X = D_A(1/2, 2) and Y = Ḋ(A). Then for all p∈ (2, ∞) and w∈ A_p/2, one has
A∈(L^p(w,Y)) (the case p=2 and w=1 is allowed as well). In particular, the solution u to (<ref>) satisfies

    A u_L^p(Ω×_+,w;E)≤ C  [w]_A_p/2^max{1/2, 1/p-2}G_L^p(Ω×_+,w;γ(H,D_A(1/2,2))),

where C only depends on E,A,p.


Note that as in the first proof of Theorem <ref>, A is sectorial of angle <π/2 on the space X.
For p=2, as in the second proof of Theorem <ref>, it suffices to prove the following variant of (<ref>)

    (∫_0^∞A e^-t A x_E^2 t)^1/2≤ C x_D_A(1/2,2).

The latter estimate is immediate from the definition of D_A(1/2,2). It remains to apply Theorem <ref>. For this (see Remark <ref><ref>) it suffices to check e^-tA_L(X,Y)≤ Ct^-1/2, which follows from

    sup_t>0t^1/2 A e^-tAx_E  ≤x_D_A(1/2,∞)≲x_D_A(1/2, 2)  ,

where we used <cit.>.





 §.§ Stochastic heat equation on ^d

Next we continue with the stochastic heat equation. We will show that using only extrapolation results for stochastic singular integrals one can deduce the stochastic maximal L^p(L^q)-regularity results in <cit.> and <cit.>. Moreover we actually obtain results with weights in time. One can check that the proof of <cit.> based on the boundedness of the H^∞-calculus of -Δ actually also gives the result with weights in time, and moreover A_q-weights in spaces could be added as well. Still we find it illustrative to show in the example below that the L^2(L^2)-case can be combined with extrapolation arguments to deduce the weighted L^p(L^q)-case for all p∈ (2, ∞) and q∈ [2, ∞).
For details on Bessel potential spaces we refer to <cit.>.
[Stochastic heat equation in Bessel-potential spaces]
Let m∈, s∈, p,q∈ (2, ∞) and w∈ A_p/2 (or p=q=2, w ≡ 1). On ^d consider

    du + (-Δ)^m u t = G  W_H,    on _+,
    
    u(0)=0,

where G∈ L^p_F(Ω×_+,w;H^s,q(^d;H)). Then the mild solution u to (<ref>) satisfies

    (-Δ)^m/2u_L^p(Ω×_+,w;H^s,q(^d))≤ C  [w]_A_p/2^max{1/2, 1/p-2}G_L^p(Ω×_+,w;H^s,q(^d;H)),

where C only depends on p,q,d.


By lifting we may assume s=0 (see <cit.>). First suppose q=2. It suffices to check Corollary <ref><ref>. Note for any f∈ L^2(^d) by Plancherel's theorem

    ∫__+∫_^d|(-Δ)^m/2 e^tΔ f(x)|^2  x  t     = ∫__+∫_^d (2π|ξ|)^2m e^-2(2π|ξ|)^2m t |f(ξ)|^2 ξ t
    
        = ∫_^d∫__+(2π|ξ|)^2m e^-2(2π|ξ|)^2m t |f(ξ)|^2  t ξ
        = 1/2∫_^d |f(ξ)|^2 ξ = 1/2f_L^2(^d)^2.

Therefore by Corollary <ref> we find the desired result for q=2.

From the extrapolation theorem <cit.> we obtain the result for w=1 and p=q∈ (2, ∞). An application of Theorem <ref> gives the required estimate for all p,q∈ (2, ∞) and w∈ A_p/2.



Next we prove a similar result on Besov spaces. For details on Besov space we refer to <cit.>.
[Stochastic heat equation in Besov spaces]
Let m∈, r∈ [2,∞), s∈, p,q∈ (2, ∞) and w∈ A_p/2 (or p=q=2, w≡ 1). On ^d consider

    du +(1-Δ)^m u t = G  W_H,    on _+,
    
    u(0)=0,

where G∈ L^p_F(Ω×_+,w;B^s_r,q(^d;H)).
Then the mild solution u to (<ref>) satisfies

    (1-Δ)^m/2u_L^p(Ω×_+,w;B^s_r,q(^d))≤ C  [w]_A_p/2^max{ 1, 1/p-2}G_L^p(Ω×_+,w;B^s_r,q(^d;H)),

where C only depends on p, q, r, s, d.



Again by lifting (see <cit.>) we may assume s := 2mθ∈ (0,2m). Let E = L^r(^d) and define

    (A,D(A)) := ((1-Δ)^m, W^2m,r(^d)).

 Then A is sectorial of angle 0 and 0∈ρ(A) on E. Since D_A(θ,q) = B^s_r,q (see <cit.>) the result follows from Theorem <ref>.


 


  * There is an inconsistency between the equations (<ref>) and (<ref>) (-Δ vs. 1-Δ). The reason to consider 1-Δ is that one has the restriction 0∈ρ(A) in Theorem <ref>. With a different proof one can also consider Example <ref>  with 1-Δ replaced by -Δ. For example one can obtain this by a real interpolation argument in Example <ref>. To avoid adaptedness problems in the interpolation argument one can first consider deterministic G and afterwards apply Proposition <ref>.

  * The results of Examples <ref> and <ref> are incomparable except if r=q=2 (see <cit.>). A similar example could be proved for Triebel–Lizorkin spaces, by using <cit.> and the boundedness of the H^∞-calculus of 1-Δ on F^s_r,q(^d), which can be proved as in <cit.> with the Mihlin multiplier theorem <cit.>. We do not see a way to prove this using just extrapolation.








 §.§ Stochastic heat equation on a wedge

Our next application is an L^p(L^q)-version of the stochastic maximal regularity result in <cit.> for the stochastic heat equation on an angular domain. The deterministic setting was considered in <cit.> and later improved in <cit.> and <cit.>. At the moment it is unclear whether the Dirichlet Laplacian -Δ on an angular domain has a bounded H^∞-calculus, and how to characterize D((-Δ)^1/2) in terms of weighted Sobolev spaces. Therefore, we cannot apply <cit.> and instead we will use <cit.> and extrapolation theory to derive L^p(L^q)-regularity results.

Let κ∈ (0,2π).
On the wedge

    D:={x∈^2: x = (rcos(φ), rsin(φ)), r>0, φ∈ (0,κ)}
 consider the stochastic heat equation:

    du -Δ u t = G  W_H,    on _+,
    
    u(0)=0.

Let q∈ [2,∞) and assume θ is such that

    (1-π/κ)q <θ<(1+π/κ)q.

Then for all p∈ (2, ∞) and w∈ A_p/2 (where p=2 and w=1 if q=2 is allowed as well) the mild solution u to (<ref>) satisfies

    u_L^p(Ω×_+,w;Ẇ^1,q(D,·^θ-2))   ≤ C G_L^p(Ω×_+,w;L^q(D,·^θ-2;H))
    u_L^p(Ω×_+,w;L^q(D,·^θ-2-q))   ≤ C G_L^p(Ω×_+,w;L^q(D,·^θ-2;H)),

where C only depends on d,p,q,[w]_A_p/2,θ and κ. Here Ẇ^1,q(D,·^θ-2) denotes the usual homogenous Sobolev space of distributions  u such that ∂_j u∈ L^q(D,·^θ-2).


In <cit.> (<ref>) was proved for p=q and w=1, where it was stated for bounded intervals (0,T). Since it holds with T-independent constants one can let T→∞ to find the result on _+. In order to prove the result for p≠ q we will use Theorem <ref> with

    X    := L^q(D,·^θ-2))
    
      Y   :=Ẇ^1,q(D,·^θ-2)∩ L^q(D,·^θ-2-q).

By Proposition <ref> -Δ is sectorial of angle <π/2 and e^t Δ_L(X,Y)≤ C t^-1/2 for t>0, so that Y is allowed in Theorem <ref> (see Remark <ref><ref>), and hence the result follows.




 §.§ Non-autonomous case with time-dependent domains

In this subsection we prove extrapolation results under the conditions introduced by Acquistapace and Terreni <cit.> (see also <cit.> and references therein). In the deterministic case extrapolation of maximal L^p-regularity was proved in <cit.> under the Acquistapace–Terreni conditions and the Kato–Tanabe condition. Here the authors consider maximal L^p-regularity on  and _+ respectively. Below we prefer to consider maximal regularity results on finite intervals (0,T) in order to avoid exponential stability assumptions. This is possible due to Section <ref> and a version of this theory could also be applied in the deterministic setting.

asmp:AT
Fix T∈ (0,∞). Next we introduce the (AT)-conditions due to Acquistapace and Terreni on a family of closed operators (A(t))_t∈ [0,T] on a Banach space X. Let us write A_w(t) = A(t) + w. We start with a uniform sectoriality condition:
[enumerate]leftmargin=40pt

  
  *  There exists a ϑ∈ (0,π/2), w≥ 0 and M>0 such that for every t∈ [0,T], one has σ(A_w(t)) ⊆Σ_ϑ and
  
    R(λ, A_w(t))_L(X)≤M/|λ|+1,   λ∈Σ_ϑ^c ,

      where
    
    Σ_ϑ := λ∈∖0: λ<ϑ.


The next condition is a Hölder continuity assumption, which depends on the change of the domains D(A(t)).


  
  *  There exist 0<μ,ν≤ 1 with μ+ν>1 and M ≥ 0 such that for all s,t∈ [0,∞) and λ∈Σ_ϑ^c,
    
    |λ|^νA_w(t)R(λ,A_w(t))(A_w(t)^-1-A_w(s)^-1)_L(X)≤ M |t-s|^μ.

[enumerate]leftmargin=25pt
When (A(t))_t∈ [0,T] satisfies both <ref> and <ref> we say that it satisfies asmp:AT(AT).

If the domains D(A(t)) all equal a fixed Banach space X_1 and

    A(t)-A(s)_L(X_1,X)≤ C  t-s^μ

for some μ>0, then (A(t))_t∈ [0,T] satisfies <ref> with ν=1. Indeed, this follows directly from the equation  A_w(t)^-1 - A_w(s)^-1 = A_w(t)^-1(A_w(s)-A_w(t))A_w(s)^-1.


The following generation result is due to Acquistapace and Terreni (see <cit.> for details).
We denote D := {(t,s) ∈ [0,T]× [0,T]: t ≥ s}.

Assume asmp:AT(AT) for (A(t))_t ∈ [0,T]. There exists a unique strongly continuous map S D→L(X) such that

    S(t,t)    = I,       t ∈ [0,T],
    
        S(t,s)S(s,r)    = S(t,r),       t≥ s≥ r,
    ddt S(t,s)    = A(t)S(t,s),        t>s.

Moreover for all 0≤α≤ 1 there exists a constant C>0
such that

    A_w(t)^α S(t,s)_L(X)   ≤ C (t-s)^-α,      (t,s) ∈ D.




Given S as in Proposition <ref>, we call (S(t,s))_t ≥ s the evolution family generated by (A(t))_t∈ [0,T]. In order to state our extrapolation result we will need some notation. For 0<α≤ 1 and t ∈ define

    X_α^t := D(A_w(t)^α).

endowed with the graph norm. Moreover set X_0^t =  D(A_w(t))^·_X. Note that since -w ∈ρ(A(t)) we have

    x_X_α^t≤ C  A_w(t)^αx_X ,      x ∈ D(A_w(t)^α).



Let 0<α≤ 1.
Let (X_β)_β∈ [0,α] be an interpolation scale and assume for β∈ [0,α] one has X_β^t ↪X_β uniformly in t ∈.
Then

    S(t,s) - I_L(X_α^s,X_β)≤ C (t-s)^α-β,     (t,s) ∈ D.



The result for β =α is clear from the assumption and <cit.>. For β = 0, the result follows from <cit.>. The result for 0<α<β follows by interpolation.


We can now prove our extrapolation theorem for (A(t))_t∈ [0,T] in the setting of Acquistapace and Terreni:
 Let α∈  (12,1] and let (X_β)_β∈ [0,α] be an interpolation scale.
Assume the following conditions:

  
  * Both (A(t))_t∈ [0,T]  and (A(t)^*)_t∈ [0,T] satisfy asmp:AT(AT).
  
  * For β∈ [0,α] one has X_β^t ↪X_β uniformly in t ∈ [0,T].
  
  * X_1/2 is a UMD Banach space with type 2

Suppose A∈(L^p(0,T;X_1/2)) for some p∈ [2, ∞). Then for all q∈ (2, ∞) and w∈ A_q/2 one has A∈(L^q((0,T),w;X_1/2)).



Set Y:= X_1/2 and let K[0,T]^2→L(X,Y) be the kernel given by

    K(t,s) = S(t,s) _t >s.

Then by our assumptions, Propositions <ref>,  <ref> and <ref> we know that K ∈K_γ(L^p(0,T)). Therefore by Theorem <ref> it suffices to check the (ϵ,2)-standard kernel conditions for K.

To do so take t>s and note that by Proposition <ref> for 0≤ s<t≤ T,

    K(t,s)_L(X,Y)≤ C A(t)^1/2 S(t,s)_L(X)≤ C(t-s)^-1/2.


To check (<ref>) on [0,T] let α∈ (1/2,1] be such that the conclusion of Lemma <ref> holds and take |t-t'|≤1/2|t-s|. If t<s, then also t'<s and there is nothing to prove. Thus it suffices to consider the case t,t'>s. If t'>t, then

    K(t',s) - K(t,s)_L(X,Y)    = K(t',t) - I_L(X_α^t,Y)K(t,s)_L(X,X_α^t)
       ≤ C  (t'-t)^α-1/2 (t-s)^-α
        = C  |t-t'/t-s|^α-1/2 |t-s|^-1/2,

where we used Lemma <ref> and Proposition <ref>.
In the case t>t' the same estimate holds with t and t' interchanged. Since
t'-s≥12(t-s), (<ref>) also follows in this case.


Next we check (<ref>). By <cit.> we have for 0≤ s<t≤ T

    S(t,s) A(s)_L(X)≤ C(t-s)^-1.

Therefore using Proposition <ref>  we have

    dds K(t,s)_L(X,Y)    = S(t,s)A(s)_L(X,Y)
       ≤ C A(t)^1/2 S(t,s) A(s)_L(X)
       ≤ C A(s)^1/2 S(s,s+t2)_L(X)S(s+t2,t) A(t)_L(X)
       ≤ C (s-t)^-3/2.

As in the proof of Lemma <ref> we obtain that (<ref>) holds with ω(r) = C r.
We can therefore conclude that K is an (α-1/2,2)-standard kernel, which finishes the proof.



  If X and X_1/2 are Hilbert spaces, the assumption that A∈(L^2(0,T;X_1/2)) in Theorem <ref> can be checked by showing

    t↦ S(t,s)x _t>s_L^2(0,T;X_1/2)≲x_X,      x∈ X,    s∈ [0,T],

using Proposition <ref><ref>. By the proof of <cit.> it is therefore sufficient to check

    t↦ A_w(s)^1/2e^t A(s)x_L^2((0,T);X)≲x,       s∈ [0,T],   x∈ X.



As an application we deduce stochastic maximal L^p-regularity for an operator family which was previously considered in <cit.> in the deterministic setting and in <cit.> and <cit.> in the stochastic setting. In particular, stochastic maximal L^2(L^2)-regularity was derived in the latter. Below we extend this to an L^p(L^q)-setting.
[Stochastic heat equation on domains with time-dependent Neumann boundary condition]
Let ϵ∈ (0,1/2) and T ∈ (0,∞). On a bounded C^3+ϵ-domain D⊆^d consider

    du +A u t = G  W_H,    on [0,T]× D,
    
    
    C u = 0    on [0,T]×∂ D,
    
     u(0)=0.

Here the differential operator A and boundary operator are given by

    A(t,x)u     = -∑_i,j=1^d ∂_i a_ij(t,x) ∂_j u,
    
     C(t,x) u     = ∑_ij=1^d a_i,j(t,x) n_i(x) ∂_j u,

where for x∈∂ D, n(x)∈^d denotes the outer normal of D. Assume that the coefficients are real-valued and satisfy

    a_ij∈ C^1+ϵ/2, 2+ϵ([0,T]×D),

for all i,j∈{1, …, d}.

We further assume (a_ij) is symmetric  and that there exists a κ>0 such that

    ∑_i,j=1^n a_ij(t,x) ξ_iξ_j≥κ|ξ|^2,   x∈ D, t∈ [0,T], ξ∈^d.

Then for all p,q∈ (2, ∞) and w∈ A_p/2 (where p=q=2 and w=1 is allowed as well) the mild solution u to (<ref>) satisfies

    u_L^p(Ω× (0,T),w;W^1,q(D))≤ C G_L^p(Ω× (0,T),w;L^q(^d;H)),

where C does not depend on G.



  Example <ref> for p=q=2 and w ≡ 1 has been shown in <cit.> assuming only a C^2 domain and
  
    a_ij   ∈ C^1/2+ϵ([0,T];C(D)), 
    
    a_ij(t,·)    ∈ C^1(D), 
    ∂_k a_ij   ∈ C([0,T]×D)

for all i,j,k∈{1, …, d} and t∈ [0,T] and some μ∈ (12,1]. This in turn can be extrapolated to p ∈ [2,∞), q=2 and w ∈ A_p/2 as in Step 2 of the following proof. However, in this situation the kernel estimates in <cit.> are not strong enough to check the parabolic Hörmander condition needed in Step 1 of the following proof. Therefore only L^p(L^2) theory can be obtained in this setting using the kernel estimates available in literature.



In <cit.> the result has been shown for p=q=2 and w ≡ 1, we will use extrapolation techniques to deduce the general case. For this note that in <cit.> it is shown that for q ∈ (1,∞) the realization of (A(t))_t ∈ [0,T] on L^q(D) with domain

    D(A(t)) := u ∈ W^2,q(D):_∂ D(C(t,·)u)=0

satisfies asmp:AT(AT).

Step 1: We will first use <cit.> to deduce the result for p=q and w≡1. For this let Γ denote the Green kernel of the evolution family associated to the realization of A on L^2(D), which exists by <cit.>. Then the mild
solution u to (<ref>) is given by

    u(t,x) = ∫_0^t ∫_D Γ(t,s,x,y) G(s,y) y  W_H(s),      (t,x) ∈ (0,T)× D,

For α= 1 define

    K_α(t,s,x,y) = ∂_x^αΓ(t,s,x,y),      t,s ∈ (0,T),    x,y ∈ D.

Then by <cit.> we have for all β=1, t>s and x,y ∈ D

    ∂^β_x K_α(t,s,x,y)   ≤ C 1/t-s^(α+d+1)/2exp-c x-y/(t-s)^1/2
    ∂_t  K_α(t,s,x,y)   ≤ C 1/(t-s)^(d+3)/2exp-c x-y/(t-s)^1/2

from which the assumption, and therefore the conclusion of Lemma <ref> follows using Lemma <ref>. If we extend K_α by zero for t,s ≥ T the same conclusion holds on _+ × D, which has infinite measure.
Combined with the case  p=q=2 from <cit.>  we have checked the assumptions of <cit.> for the operators

    T_α L^2(Ω×_+;L^2(^d;H)) → L^2(Ω×_+;W^1,2(D))

given by

    T_α G(t,x) := ∫_0^t ∫_D K_α(t,s,x,y) G(s,y) y  W_H(s),    (t,x) ∈_+× D,

for all α = 1 and thus the result for p=q and w≡1 follows.

Step 2: For the general case let X_β = W^2β,q(D) for β∈ (0,1] and X_0 = L^q(D). Then X_β^t ↪X_β for all β∈ [0,1] (see <cit.>) and by Step 1 we have A∈(L^q(0,T;X_1/2)). Therefore the result in the general case follows from Theorem <ref>.






 §.§ Volterra equations


In <cit.> the results of <cit.> have been extended to the setting of integral equations: 
    U(t) +  A∫_0^t 1/Γ(α) (t-s)^α-1 U(s)  s =  ∫_0^t 1/Γ(β) (t-s)^β-1 U(s)  W_H(s),

where α∈ (0,2) and, β∈ (12,2).  The solution U is given by

    U(t) = ∫_0^t S_αβ(t-s) G(s)   W_H(s),      t ∈_+,

where S_αβ is the so-called resolvent associated with A, α and β.
The maximal regularity result in <cit.> gives L^p-estimates for A^θ∂^η_t U in terms of G, where β-αθ-η = 1/2 with θ∈ [0,1) and η∈ (-1,1). In this case one has to estimate a stochastic convolution with kernel k(t) = A^θ∂^η_t S_αβ(t). We will not go into details on Volterra equations further now, but restrict ourselves to checking that K(s,t):=k(s-t) is an (ϵ, 2)-standard kernel for suitable ϵ∈ (0,12). Consequently our extrapolation theorem can be applied to this setting as well.

First consider η∈(-12,1). Choose ϵ∈ (0,12) such that η+ϵ∈ (0,1), then there is an M>0 such that (see <cit.>)

    ∂^ϵ k(t)≤ M  t^-ϵ-1/2.

Writing K(t) = ∂^-ϵ∂^ϵ  k(t), it follows from Lemma <ref> that K is an (ϵ, 2)-standard kernel.

If η∈(-1,-12), we let ϵ = -η. Then k(t) = ∂^-ϵ_t  A^θ S_αβ(t) and there is an M>0 such that (see <cit.>)

    A^θ S_αβ(t)≤ M t^-1/2-ϵ

Therefore, K is an (ϵ, 2)-standard kernel.




§ P-INDEPENDENCE OF THE R-BOUNDEDNESS OF CONVOLUTIONS

In this final section we prove the p-independence of a Banach space property which was introduced in <cit.>. In order to state the condition we need to introduce the notion R-boundedness of a family of operators.

For details on R-boundedness we refer to <cit.>. For us it will be enough to recall the definition. Let X and Y be Banach spaces and let (ε_j)_j≥ 1 be a Rademacher sequence on a probability sequence (Ω, 𝒜,). A family of operators T⊆L(X,Y) is called R-bounded if there exists a constant C such that for all T_1, …, T_n∈𝒯 and x_1, …, x_n∈ X one has

    ∑_j=1^n ε_j T_j x_j_L^2(Ω;X)≤ C∑_j=1^n ε_jx_j_L^2(Ω;X).



Let X be a Banach space with type 2.
For λ∈ with (λ)>0 let k_λ:_+→ be given by

    k_λ(s) = λ^1/2 e^-λ s,      s ∈_+,

and define T_λ:L^p(_+;X)→ L^p(_+;γ(_+;X)) by

    T_λ f(s) = k_λ(s-·) f(·),      s∈_+.

Then by Proposition <ref>

    k_𝒦_γ(L^p(_+))≤τ_2,X(|λ|/2(λ))^1/2.

The following p-dependent condition was introduced in <cit.>.
[enumerate]leftmargin=40ptasmp:Cp


  *  For each θ∈ [0,π/2) the family 𝒯 = {T_λ: |(λ)|≤θ} is R-bounded
from L^p(_+;X) into L^p(_+;γ(_+;X)).

[enumerate]leftmargin=25pt

Note that (<ref>) implies that 𝒯 is uniformly bounded.
In <cit.> the condition <ref> was combined with the boundedness of the H^∞-calculus in order to derive stochastic maximal L^p-regularity (see Definition <ref>).

From  <cit.> it can be seen
that in the following case the condition <ref> holds for all p∈ (2, ∞):


  * X is a 2-convex Banach function space and the dual of its concavification X^2 is an -space, i.e. the lattice Hardy–Littlewood maximal operator is bounded on L^p(^d;(X^2)^*) for some (all) p ∈ (1,∞).

In particular, UMD Banach function spaces are HL-spaces, but also L^∞ is an -space. In particular, the space L^q satisfies <ref> for any q∈ [2, ∞) and p∈ (2, ∞). In the case q=2 one can additionally allow p=2. On the other hand, L^q for q>2 fails asmp:Cp(C_2) (see <cit.> and the proof of <cit.>). A Banach function space with UMD and type 2 for which we do not know whether <ref> holds for p∈ (2, ∞) is for instance ℓ^2(ℓ^4). Some evidence against this can be found in <cit.>.

It was an open problem whether <ref> is p-independent. Below we settle this issue. In the special case of Banach function spaces one could also derive this by rewriting <ref> as a square function result (see <cit.>) and using operator-valued Calderón–Zygmund theory (see <cit.>).

Let X be Banach space with type 2  and let p∈ [2, ∞). If <ref> holds,
then for all θ∈ [0,π/2), q∈ (2, ∞) and  w∈ A_q/2(_+) the family 
    𝒯 = {T_λ: |(λ)|≤θ}
 is R-bounded
from L^q(_+,w;X) into L^q(_+,w;γ(_+;X)).
In particular asmp:Cp(C_q) holds for all q∈ (2, ∞).



Fix n∈. Let λ_1, …, λ_n∈Σ_θ and f_1, …, f_n∈ L^q(_+,w;γ(_+;X)). Let _n(X) be the space X^n endowed with the norm

    (x_j)_j=1^n__n(X) := ∑_j=1^n ε_jx_j_L^2(Ω;X),

where (ε_j)_j=1^n is a Rademacher sequence. Replacing the L^2(Ω;X)-norm by L^r(Ω;X) with r∈ [1, ∞) leads to an equivalent norm by the Kahane–Khintchine inequalities (see <cit.>).
Define a diagonal operator k:_+→L(_n(X)) by

    (k(s)x)_j = k_λ_j(s) x_j,      j∈{1, …, n},    x ∈_n(X),

and set K(s,t) := k(s-t). To prove the required R-boundedness of 𝒯, by the Kahane–Khintchine inequalities, Fubini's theorem and Proposition <ref> it suffices to prove that
K_𝒦_γ(L^q(_+,w))≤ C where C is independent of n. Now by <ref> we know the latter is true for w=1 and q=p. Therefore, by Theorem <ref> (see also Section <ref>) it suffices to check that K satisfies the required Dini condition with constants only depending on θ. For this we check the condition of Lemma <ref>. Moreover, since K is of convolution type it suffices to check that K'(s)≤ C s^-3/2. Since k'(s) is a diagonal operator we have for x ∈_n(X):

    K'(s)x__n(X) = ∑_j=1^n ε_j k_λ_j'(s) x_j_L^2(Ω;X)≤ C s^-3/2∑_j=1^n ε_j  x_j_L^2(Ω;X),

where we used the Kahane contraction principle (see <cit.>) together with 
    s^3 k_λ_j'(s)^2≤sup_λ∈Σ_θ |λ|^3 e^-2(λ)≤27/8e^3 cos^3(θ) := C^2.

This implies the required estimates for K and therefore finishes the proof.



One could replace k_λ by a class of functions which satisfies the (ω,2)-Dini condition uniformly for one fixed function ω. Moreover, a similar result holds on other spaces of homogenous type.






§ TECHNICAL ESTIMATES




 §.§ Heat kernel estimates on a wedge


Assume κ∈ (0,2π), q∈ (1, ∞), σ>0 and θ∈.
Let μ = π/κ∈ (12,∞). For j∈{0,1,2} let k_t^j:^2→ [0,∞) be defined by

    k_t^j(x,y) = ζ^μ-j(t,x) ζ^μ(t,y) t^-1exp(-σ|x-y|^2/t),

where ζ(t,x) = |x|/|x|+√(t). For j-μ<θ/q<2+μ one has

    sup_t>0,y∈^2 ∫_^2 k_t^j(x,y) |x|^θ/q |y|^2-θ/q d x/|x|^2   <∞,
    
    sup_t>0,x∈^2 ∫_^2 k_t^j(x,y) |x|^θ/q |y|^2-θ/q d y/|y|^2   <∞.



By a substitution replacing x and y by x √(t) and y √(t), one can check that it suffices to consider t=1, and we set k^j(x,y) = k_1^j(x,y).
It suffices to consider σ∈ (0,1]. Moreover, since ζ(1,x) ≤ζ(1,x/σ) ≤1/σζ(1,x), by a substitution one can reduce to σ=1. Let a = μ+2-θ/q. Then a>0 by the assumptions in the lemma, and a simple rewriting shows that

    k^j(x,y) |x|^θ/q |y|^2-θ/q  = |x|^2μ+2-j-a/(|x|+1)^μ-j|y|^a/(|y|+1)^μ e^-|x-y|^2 .


Step 1: First consider the integral with respect to x. One has

    ∫_^2 k^j(x,y) |x|^θ/q |y|^2-θ/q d x/|x|^2   = ∫_^2|x|^2μ-j/(|x|+1)^μ-j(|y|+1)^μy/x^a^-|x-y|^2 x
    
        = S_1 + S_2 + S_3,

where S_1 is the integral over |x|≤12|y|,  S_2 is the integral over 12|y|<|x|< 32|y| and S_3 is the integral over |x|≥32|y|.

For S_1 note that |x-y|≥ |y| - |x| ≥12 |y|. Therefore, e^-|x-y|^2≤^-1/4 |y|^2 and we find

    S_1    ≤ |y|^a ^-1/4 |y|^2∫_|x|≤1/2 |y| |x|^2μ-j-a (|x|+1)^j-μ x
    
       ≤ 2π (|y|+1)^|j-μ|+a^-1/4 |y|^2∫_0^1/2 |y| r^2μ-j-a+1 r
    
        (|y|+1)^|j-μ|+a  |y|^2μ-j-a+2^-1/4 |y|^2≤ C,

where we used 2μ-j-a+2 = μ-j+θ/q>0.

For S_2 if |y|≤1, then

    S_2    ≤∫_1/2|y|<|x|< 3/2|y|(|x|/|x|+1)^μ-j(|x|/|y|+1)^μ x
    
       ∫_1/2|y|<|x|< 3/2|y| |x|^2μ-j x |y|^2μ-j+1≤ C,

where we used 2μ-j+1>2-j≥ 0.
If |y|>1, then S_1  ∫_^2 e^-|x-y|^2 x = C.

For S_3, note that |x-y|≥ |x|-|y|≥1/3 |x|. Thus e^-|x-y|≤ e^-1/9 |x|^2. Now if |y|>1, then

    S_3    ≤ |y|^a-μ∫_|x| > 3/2|y| |x|^μ-a^-1/9 |x|^2 x 
        = 2π |y|^a-μ∫_3/2|y|^∞ r^μ-a+1^-1/9 r^2 r
    
        = 2π∫_3/2^∞ s^μ-a+1 |y|^2 ^-1/9 |y|^2 s^2 s
    
       ≤  2π |y|^2 ^-1/36 |y|^2∫_3/2^∞ s^μ-a+1^-1/36 |s|^2 s ≤ C,

where we used |y|s ≥12(|y|+s) for s,y>1.
If |y|≤ 1, then

    S_3    ≤∫_|x| > 3/2|y| |x|^2μ-j-a (1+|x|)^j-μ^-1/9 |x|^2 x
    
       ≤ 2π∫_0^∞ r^2μ-j-a+1 (1+r)^μ-j^-1/9 r^2 x <∞,

because 2μ-j-a+2 = μ-j+θ/q>0.


Step 2: Next consider the integral with respect to y. One has

    ∫_^2 k(x,y) |x|^θ/q |y|^2-θ/q d y/|y|^2   = ∫_^2|x|^2μ-j/(|x|+1)^μ-j(|y|+1)^μy/x^a-2^-|x-y|^2 y
    
        = T_1 + T_2 + T_3,

where T_1 is the integral over |y|≤12|x|,  T_2 is the integral over 12|x|<|y|< 32|x| and T_3 is the integral over |y|≥32|x|.

For T_1 note that |x-y|≥ |x| - |y| ≥12 |x|. Therefore, e^-|x-y|^2≤ e^-1/4 |x|^2 and we find

    T_1    ≤ |x|^2μ+2-a-j (|x|+1)^|μ-j|^-1/4 |x|^2∫_|y|≤1/2 |x| |y|^a-2 y
    
        = 2π |x|^2μ+2-a-j (|x|+1)^|μ-j|^-1/4 |x|^2∫_0^1/2 |x| r^a-1 r
    
        |x|^2μ+2-j (|x|+1)^|μ-j|^-1/4 |x|^2≤ C,

where we used a>0 and 2μ+2-j>0.

For T_2 if |x|≤ 1 then we can write

    T_2   ≲(|x|/|x|+1)^2μ-j∫_1/2|x|<|y|< 3/2|x|^-|x-y|^2 y ≲  |x|^2μ-j+2≤ C.

where we used 2μ+2-j> 0. If |x|≥ 1, then

    T_2≲∫_^2 e^-|x-y|^2 y = C.




For T_3, note that |x-y|≥ |y|-|x|≥1/3 |y|. Thus e^-|x-y|≤ e^-1/9 |y|^2.
If |x|>1 we can write

    T_3    ≲∫_|y|>3/2|x|(|y|/|x|)^a-2-μ e^-1/9|y|^2 y
    
        = 2π∫_3/2|x|^∞(r/|x|)^a-2-μ e^-1/9r^2 r  r
    
        = 2π∫_3/2^∞ s^a-1-μ |x|^2 ^-1/9 |x|^2 s^2 s
    
       ≤ 2π |x|^2 e^-1/36 |x|^2∫_3/2^∞ s^a-1-μ^-1/36 s^2 s≤ C.

If |x|≤ 1, then since 2μ-j-a+2≥ 0,

    T_3    ≲∫_|y|>3/2|x| |y|^a-2^-1/9|y|^2 y ≤ 2π∫_0^∞ r^a-1 e^-1/9r^2 r <∞.

This finishes the proof.


Let κ∈ (0,2π). On the wedge

    D:={x∈^2: x = (rcos(φ), rsin(φ)), r>0, φ∈ (0,κ)}
 consider heat equation:

    u_t= Δ u,    on _+× D,
    
    u(0,y)=f(y).

Let Γ denote the Green kernel of the heat semigroup associated to (<ref>). The solution to (<ref>) is given by  (see <cit.>)

    e^t Δ f(x) = ∫_DΓ(x,y,t) f(y)  y.

In the next proposition we collect some properties of the heat semigroup on the wedge D.

Assume κ∈ (0,2π), q∈ (1, ∞), θ∈ and set

    X    = L^q(D,·^θ-2),
    
      Y    =Ẇ^1,q(D,·^θ-2)∩ L^q(D,·^θ-2-q)

The following assertions hold:


  *  If -π/κ<θ/q<2+π/κ, then -Δ is a sectorial operator of angle <π/2 on X. In particular, (e^tΔ)_t≥ 0 is a bounded analytic semigroup on L^q(D,·^θ-2);

  *   If 1-π/κ<θ/q<2+π/κ, then sup_t>0 t^1/2e^tΔ_L(X,Y)<∞.


Although -Δ is sectorial of angle <π/2 for a large range of values of θ, we do not know its domain on the full range.
Although we do not need it we note that if 2-π/κ<θ/q<2+π/κ, then (see <cit.>)

    D(Δ) = u: u,u/·^2, ∂^α u∈ L^q(D,·^θ-2)  for  |α|=2.

The domain for other values of θ seems more difficult to characterize.


Let μ = π/κ∈ (1/2,∞). For <ref> first suppose 2-μ<θ/q<2+μ. It follows from <cit.> that -Δ has deterministic maximal regularity. Thus in this case  <ref>  follows from <cit.>.
 The range -μ<θ/q<μ follows by a duality argument from the range  2-μ<θ/q<2+μ, since

    L^q(D,·^θ-2)^* = L^q'(D,·^θ-2)

with θ = (2q-θ)/(q-1). The remaining range μ≤θ/q≤ 2-μ follows by complex interpolation (see <cit.>).

For <ref> we use the following estimates for Γ (see <cit.>):

    |∂_x^αΓ(x,y,t)| ≤ Cζ^μ-|α|(t,x) ζ^μ(t,y) t^-2+|α|/2exp(-σ|x-y|^2/t),    α≤ 1

where ζ(t,x) = |x|/|x|+√(t). Therefore it suffices to prove for f ∈ L^q(D,·^θ-2)

    x↦∫_D k_t(x,y) f(y)  y_L^q(D,·^θ-2)≤ Cf_L^q(D,·^θ-2),

where k_t(x,y) is either

    ζ^μ-1(t,x) ζ^μ(t,y) t^-1exp(-σ|x-y|^2/t),    or   ζ^μ(t,x)  ζ^μ(t,y) x^-1  t^-1/2exp(-σ|x-y|^2/t),

where (<ref>) and (<ref>) correspond  to the boundedness in  Ẇ^1,q(D,|x|^θ-2) and L^q(D,|x|^θ-2-q) respectively. Since (<ref>)≤(<ref>)  it suffices to prove the boundedness for the case (<ref>).
A simple rewriting shows that it is enough to prove for g ∈ L^q(D,·^-2)

    x↦∫_D k_t(x,y) |x|^θ/q |y|^2-θ/q g(y)  d y/y^2_L^q(D,·^-2)≤ C g_L^q(D,·^-2),

To prove the latter by Schur's lemma (see <cit.>) it suffices to show

    sup_t>0,y∈^2 ∫_D k_t(x,y) |x|^θ/q |y|^2-θ/q d x/|x|^2   <∞,
    
    sup_t>0,x∈^2 ∫_D k_t(x,y) |x|^θ/q |y|^2-θ/q d y/|y|^2   <∞,

which follows from Lemma <ref>.




 §.§ Parabolic Hormänder and Dini conditions on a smooth bounded domain

Define the parabolic norm on ×^d by

    (t,x)_(2,1) := maxt^1/2, x,      (t,x), ∈×^d.

Let D ⊆^d be a smooth bounded domain and fix T ∈ (0,∞]. We equip (0,T) × D with the parabolic metric induced by  ·_(2,1), which turns it into a space of homogeneous type (see also Section <ref>). We will show that versions of Lemma <ref> and Lemma <ref> work in this setting.


  Fix T ∈ (0,∞] and let
  K(0,T)× (0,T) × D × D →
 be measurable such that K(t,s,x,y)=0 for t<s. Suppose there exist C,c>0 such that for α≤ 1,
 
    ∂^α_x K(t,s,x,y)   ≤ C 1/(t-s)^(α+d+1)/2 exp-cx-y/(t-s)^1/2,
    ∂_t K(t,s,x,y)   ≤ C 1/(t-s)^(d+3)/2 exp-cx-y/(t-s)^1/2,

for all t>s and x,y ∈ D. Then
 
    Kt,s,x,y- Kt',s,x',y≲ (t-t',x-x')_(2,1)/(t-s,x-y)_(2,1)^d+2

for all (t-t',x-x')_(2,1)≤1/2(t-s,x-y)_(2,1).




Take (t-t',x-x')_(2,1)≤1/2(t-s,x-y)_(2,1) and define

    [15pt]A   := Kt,s,x,y-Kt',s,x,y,
    [15pt]B   := Kt',s,x,y-Kt',s,x',y.

By the triangle inequality it suffices to estimate [15pt]A and [15pt]B separately. Let us first consider [15pt]A. If t,t'<s there is nothing to prove. If t'<s<t we have t-t'>t-s and thus also 1/2x-y^2 ≥ t-t'. Therefore using (<ref>) with α=0 we have the estimate

    [15pt]A   ≲t-s^1/2/t-s^(d+2)/2exp-cx-y/(t-s)^1/2≲(t-t',x-x')_(2,1)/t-s,x-y_(2,1)^d+2.

If s<t'<t we first consider the case that x-y≤t-s^1/2. Then by (<ref>) and the fundamental theorem of calculus we have

    [15pt]A   ≲t-t'/t'-s^(d+3)/2≤ 2 (t-t',x-x')_(2,1)^2/t-s,x-y_(2,1)^d+3,

since in this case

    t'-s≥t-s - t-t'≥12 t-s.

Next if t-s^1/2≤x-y, then again by (<ref>) and the fundamental theorem of calculus we have

    [15pt]A   ≲t-t'/x-y^d+3≤(t-t',x-x')_(2,1)^2/t-s,x-y_(2,1)^d+3

The cases t<s<t' and s<t<t' are treated similarly with the roles of t and t' interchanged.

Now for [15pt]B suppose that t'>s. Let γ:[0,1]→ D be a smooth curve from x to x' such that

    sup_r ∈ [0,1]γ'(r)    ≲x-x'
    inf_r ∈ [0,1]γ(r)-y   ≳minx-y,x'-y,

which exists since D is smooth and bounded.
 We first consider the case that x-y≤t-s^1/2. Then by (<ref>) and the fundamental theorem of calculus we have

    [15pt]B   = ∫_0^1 / r Kt,s,γ(r),y r
       ≲x-x'/t'-s^(d+2)/2≤ 2 (t-t',x-x')_(2,1)/(t-s,x-y)_(2,1)^d+2,

since (<ref>) is valid in this case. Similarly if t-s^1/2≤x-y we have

    [15pt]B   ≲x-x'/minx-y,x'-y^d+2≤ 2 (t-t',x-x')_(2,1)/t-s,x-y_(2,1)^d+2,

since in this case

    x'-y≥x-y - x-x'≥12 x-y.



Now we give an analog of Lemma <ref> in the parabolic setting:

  Fix T ∈ (0,∞], let
  K(0,T)× (0,T) × D × D →
 be measurable  and take ϵ∈ (0,1]. Suppose there is an A_0>0 such that
 
    Kt,s,x,y-Kt',s,x',y≤ A_0  (t-t',x-x')^ϵ_(2,1)/(t-s,x-y)_(2,1)^ϵ+d+1

for all (t-t',x-x')_(2,1)≤1/2(t-s,x-y)_(2,1). Then for all (t,x),(t',x') ∈ (0,T)× D we have

    ∫_0^T ∫_I(s)Kt,s,x,y-Kt',s,x',y y^2 s^1/2≤ C_ϵ  A_0

where

    I(s) := y ∈ D: (t-t',x-x')_(2,1)≤12 (t-s,x-y)_(2,1).




Take (t,x),(t',x') ∈ (0,T)× D and define ρ = (t-t',x-x')_(2,1). Then by assumption we have

    ∫_0^T ∫_I(s)   Kt,s,x,y-Kt',s,x',y y^2 s^1/2
       ≤ A_0 ∫_0^T ∫_I(s)(t-t',x-x')^ϵ_(2,1)/(t-s,x-y)_(2,1)^ϵ+d+1 y^2 s^1/2
       ≤ A_0 ∫_∫_I_1(r)∪ I_2(r)ρ^ϵ/(r,z)_(2,1)^ϵ+d+1 z^2 r^1/2
       ≤ A_0 (S_1+S_2)

where S_1 and S_2 are the parts of the inner integral over I_1(r) and I_2(r) respectively with

    I_1(r)    := z ∈ D:  (r,z)_(2,1)≥ 2ρ and (r,z)_(2,1) = z
    
     I_2(r)    := z ∈ D:  (r,z)_(2,1)≥ 2ρ and (r,z)_(2,1) = r^1/2

For S_1 we have
   
    S_1    = r ↦∫_I_1(r)ρ^ϵ/z^ϵ+d+1 z_L^2()
       ≲r ↦∫_max2ρ,r^1/2^∞ρ^ϵ/u^ϵ+2 u_L^2()
       ≲r ↦∫_2ρ^∞ρ^ϵ/u^ϵ+2 u_L^2(0,4ρ^2)+ r ↦∫_r^1/2^∞ρ^ϵ/u^ϵ+2 u_L^2(4ρ^2,∞)
       ≲ρ^-1∫_0^4ρ^2 r^1/2 + ρ^ϵ∫_4ρ^2^∞1/r^ϵ+1 r^1/2≤ C_ϵ,

   and for S_2 we have
   
    S_2    ≤ 2 r ↦∫_z≤ r^1/2ρ^ϵ/r^(ϵ+d+1)/2 z_L^2(4ρ^2,∞)
       ≲ρ^ϵr ↦1/r^(ϵ+1)/2_L^2(4ρ^2,∞)≤ C_ϵ,

   which proves the lemma.
  


alpha-sort



