The Kato Square Root Problem for Divergence Form Operators with Potential
Julian Bailey



The Kato square root problem for
divergence form elliptic operators with potential V : ^n→ is the
equivalence statement L + V^1/2 u_2≃∇ u_2 +
V^1/2 u_2, where 
L + V := - div (A ∇) + V and the
perturbation A is an L^∞
complex matrix-valued function satisfying an accretivity
condition. This relation is proved for any potential with range
contained in some positive sector and satisfying
V^α/2 u_2 + (-Δ)^α/2u_2≲V - Δ^α/2u_2 for all u ∈
D(V - Δ) and some α∈ (1,2]. The class of
potentials that will satisfy such a condition is known to contain the reverse Hölder
class RH_2 and
L^n/2^n in dimension n > 4. To prove the Kato estimate with potential, a non-homogeneous version of the framework introduced by A. Axelsson,
S. Keith and A. McIntosh for proving quadratic estimates is
developed. In addition to applying this non-homogeneous framework to the
scalar Kato problem with zero-order potential, it will also
be applied to the Kato
problem for systems of equations with
zero-order potential.


Two Algorithms for the Package-Exchange Robot-Routing Problem
    James Drain
    written June 26, 2015
=============================================================





§ INTRODUCTION




Key words and phrases.
  Kato problem; non-homogeneous; Schrödinger operator; divergence form operator;
  potential; quadratic estimates. 

  Mathematics Subject Classification. Primary 42B37 · Secondary
  35J10 

This research was partially supported by the Australian
Research Council through the Discovery Project DP160100941.


For Hilbert spaces ℋ and 𝒦, let ℒℋ, 𝒦 denote the
space of bounded linear operators from ℋ to 𝒦
and set ℒℋ := ℒℋ, ℋ. Fix
n ∈^* = ∖ 0 and let A ∈ L^∞^n; ℒ^n. Consider the sesquilinear form 𝔩^A :
H^1^n H^1^n→ defined by

    𝔩^Au,v := ∫_^n⟨  A(x) ∇ u (x) , ∇
    v(x) ⟩_^n  dx

for u,   v ∈ H^1^n. Suppose that 𝔩^A
satisfies the Gårding inequality

    Re𝔩^Au,u≥κ^A∇ u^2_2

for all u ∈ H^1^n, for some κ^A > 0. A well-known
representation theorem from classical form theory (c.f. <cit.>)
asserts the existence of an associated operator L : D
L⊂ L^2^n→ L^2^n 
for which

    𝔩^Au,v = ⟨ L u, v ⟩_2

for all v ∈ H^1^n and u in the domain of
L,

    D L =  u ∈ H^1^n : ∃  w ∈
    L^2^n s.t.  𝔩^Au,v = ⟨ w , v
    ⟩_2 ∀ v ∈ H^1^n.

The operator L  is a densely defined maximal accretive operator that
is denoted

    L = - div( A ∇).

 Due to the accretivity of the operator, it is possible to define a square root operator
√(L), with core D L, that satisfies 
√(L)·√(L) =
L. The Kato square root problem asks: what is the domain of definition
of the square root operator? This problem, first posed by Tosio Kato over 50
years ago, was conjectured to have the following solution.


 
  
 The domain of √(L) is
 D(√(L)) = H^1^n
and for any u ∈ H^1^n

    √(L) u_2≃∇ u_2.

 


 This long-standing problem withstood
solution until 2002 where it was proved using local T(b) methods by Steve Hofmann, Michael
Lacey and Alan McIntosh in <cit.> under the additional assumption of Gaussian
heat kernel bounds and in full generality by
Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh and
Phillipe Tchamitchian in  <cit.>.
 We will be interested in an alternate method of proof that was
built from similar principles and appeared a few years later.



Let Π := Γ + Γ^* be a Dirac-type operator on a Hilbert
space ℋ and Π_B
:= Γ + B_1Γ^* B_2 be a perturbation of Π by
bounded operators B_1 and B_2. Typically, Π is considered
to be a first-order system acting on ℋ := L^2^n;
^N for some n,   N ∈^* and the perturbations B_1 and
B_2 are multiplication by matrix-valued functions B_1,   B_2∈ L^∞^n; ℒ^N. In their seminal paper
<cit.>, A. Axelsson, S. Keith and A. McIntosh
developed a general framework for proving that the perturbed operator
Π_B possessed a bounded holomorphic functional calculus. This
ultimately amounted to obtaining square function estimates of
the form

    ∫^∞_0Q^B_t u^2dt/t≃u^2,

where Q^B_t := t Π_BI + t^2Π_B^2^-1 and u is
contained in the range R Π_B. They
proved that this estimate would follow entirely from a set of simple
conditions imposed upon the operators Γ, B_1 and B_2,
labelled (H1) - (H8). Then, by checking this list of simple
conditions, the Axelsson-Keith-McIntosh framework, or AKM framework
by way of abbreviation, could be used to conclude that the
particular selection of operators

    Γ := [ 0 0; ∇ 0 ],    B_1 = I,    B_2 = [ I 0; 0 A ],

defined on L^2^n⊕ L^2^n;^n, would
satisfy (<ref>) and therefore Π_B would possess a bounded
holomorphic functional calculus. The Kato square root estimate then
followed almost trivially from this.







Many classical problems from harmonic analysis will have a direct
counterpart in the Schrödinger operator setting. Adhering with
this theme, one can consider the Kato square root problem with
potential. Let V : ^n→ be a measurable function
that is contained in L^1_loc^n. 
V can be
viewed as a densely defined closed multiplication operator on
L^2^n with domain

    D V =  u ∈ L^2^n : V · u ∈ L^2^n.

The density of D(V) follows from the condition V ∈ L^1_loc(^n). 
Define the subspace

    H^1,V^n := H^1^n∩ D
    V^1/2 :=  u ∈
    H^1^n : V^1/2· u ∈ L^2^n.

Here
the complex square root V^1/2 is defined via the principal
branch cut along the negative real axis. It is easy to see that
H^1,V(^n) is dense in L^2^n since
C^∞_0(^n) ⊂ H^1,V(^n).

Let A ∈ L^∞^n;ℒ^n be as before with
(<ref>) satisfied for some κ^A > 0. Consider the sesquilinear form 𝔩_V^A : H^1,V^n H^1,V^n→
defined through

    𝔩_V^Au,v := 𝔩^Au,v + ∫_^n  V(x)
    u(x) v(x)    dx

for u,   v ∈ H^1,V^n. Suppose that there exists
some κ_V^A > 0 for which
𝔩_V^A satisfies the associated Gårding inequality

    Re𝔩_V^Au,u≥κ_V^A V^1/2 u_2^2 + ∇ u_2^2

for all u ∈ H^1,V^n. 


If the range of V is contained in some sector

    S_μ^+ :=  z ∈∪∞ :
    argz≤μ or  z = 0,  ∞

for some μ∈ [0,π/2), then (<ref>) will follow
automatically from (<ref>).


Once again, the accretivity of
𝔩_V^A  implies the existence of a maximal accretive operator
associated with this form denoted by

    L + V = - div (A ∇) + V

and defined on

    D(L + V) =  u ∈ H^1,V(^n) : ∃  w ∈ L^2(^n) s.t.  𝔩^A_V[u,v] = ⟨ w, v ⟩_2 ∀ v
    ∈ H^1,V(^n) .


For
  α∈ [1,2], define 𝒲_α to be the
  class of all V ∈ L^1_loc(^n) for which 
  
    V_α := sup_u ∈ D(V - Δ)V^α/2
        u_2 + - Δ^α/2u_2/V - Δ^α/2 u_2 < ∞.

As will be proved in Section <ref>, the collection of
potential classes 𝒲_α_α∈ [1,2] is
decreasing. The largest class 𝒲_1 consists of all
locally integrable potentials with no additional restrictions and the
smallest class 𝒲_2 contains RH_2 in any dimension
and L^n/2(^n) in dimension n > 4.
In this paper, our aim is to prove the
potential dependent Kato estimate as presented in the following
theorem.



  
Let V ∈𝒲_α for some α∈ (1,2] and A ∈ L^∞^n; ℒ^n. Suppose that the Gårding inequalities
(<ref>) and (<ref>) are both satisfied
with constants κ^A > 0 and κ_V^A > 0
respectively. There exists a constant C_V > 0 such that
 
    KP
    C_V^-1 V^1/2 u_2 + ∇ u_2≤√(L + V) u_2≤ C_VV^1/2 u_2 + ∇ u_2

for all u ∈ H^1,V^n. Moreover, the constant
C_V depends on V and α through

    C_V = C̃_Vα - 1^-1 (1 + V_α^2),

where C̃_V only depends on V through κ_V^A and
is independent of α.


This theorem is actually a statement concerning the domain of the
square root operator √(L + V). Indeed,
(<ref>) implies the equality

    D(√(L + V)) = H^1,V(^n).


  
  The above theorem tells us, in particular, that the Kato estimate
 with potential, (<ref>), is valid for any
 potential V with range contained in S_μ+ for some μ∈
 [0,π/2) and such that V is either contained in RH_2 for any
 dimension or in L^n/2(^n) in dimension n > 4.
  In
 <cit.>, it is suggested but not
 proved
 that V_α < ∞ will occur
 provided that V∈ RH_q with q > α. If this is
 indeed the case then Theorem <ref> will imply that
 the Kato estimate is true for any potential
 V with range contained in S_μ+ for some μ∈
 [0,π/2) and
 with V contained in RH_1 = A_∞.

 
In direct analogy to the potential free case, the Kato problem with
potential will be solved by constructing appropriate potential
dependent Dirac-type operators and demonstrating that they retain
a bounded holomorphic functional calculus under perturbation. In
particular, this strategy will be applied to the Dirac-type operator

    Π_V^1/2 := Γ_V^1/2 + Γ_V^1/2^* := [     0     0     0; V^1/2     0     0;     ∇     0     0 ] + [     0 V^1/2 - div;     0     0     0;     0     0     0 ]

defined on L^2(^n) ⊕ L^2^n⊕ L^2^n ; ^n, under the perturbation

    B_1 = I,    B_2 := [          I          0          0;          0 e^i ·arg V          0;          0          0          A ].

It should be observed that the operator
Γ_V^1/2 is not first-order homogeneous due to
the presence of the zero-order potential term. It will therefore not
necessarily satisfy the two conditions of the original AKM framework that
are intended to capture the first-order homogeneity property, the cancellation and coercivity conditions, (H7)
and (H8). As such, the original AKM framework cannot be directly applied. The key
difficulty in proving our result is then to alter
the original framework in order to allow for such operators. In
particular, a non-homogeneous version of the Axelsson-Keith-McIntosh
framework will be developed to handle operators of the form

    Γ_J := [ 0 0 0; J 0 0; D 0 0 ],

where D is a homogeneous first-order differential operator and J
is a possibly non-homogeneous differential operator of order less than
or equal to one. The technical challenge presented
by the inclusion of the non-homogeneous part J will be overcome by separating our
square function norm into components and demonstrating that the
non-homogeneous term will allow for the first two components to be bounded
while the third component can be bound using an argument 
similar to the classical argument of <cit.>.

Since the operator Γ_J is of a more general form than
Γ_V^1/2, the non-homogeneous AKM framework that we develop will
have applications not confined to zero-order scalar
potentials. Indeed, the non-homogeneous framework will also be used to
prove Kato estimates for systems of equations with
zero-order potential and for scalar equations with first-order
potentials. 


The structure of this paper is as follows. Section
<ref> is quite classical in nature. It provides a brief survey of
the natural functional calculus for bisectorial operators. It also
describes how
to prove the boundedness of holomorphic functional calculus from
square function estimates for a bisectorial operator. These classical proofs are repeated
here to track the dependence of the estimates on relevant
constants. This will be required to determine the potential dependence
of the constants in
Theorem <ref>. Section <ref>
describes the non-homogeneous AKM framework and
states the main results associated with it. Section <ref> contains most of the technical
machinery and is dedicated to a proof of our main result. Section
<ref> will apply the non-homogeneous AKM framework to
the scalar Kato problem with potential, the Kato problem for systems
with zero-order potential and the scalar Kato problem with first-order
potential. It is here that a proof of Theorem <ref>
will be completed. 
Finally, in Section <ref>, we will provide a meta-discussion
on the proof techniques used and compare our work with what has been
previously accomplished on non-homogeneous Kato-type
estimates. 







 §.§ Acknowledgements
 This paper is part of my PhD thesis
undertaken at the Australian National University. I am very thankful to my 
supervisor Pierre Portal for his numerous suggestions and corrections
and for the encouragement that made this article possible. I am
indebted to El Maati Ouhabaz for the discussions surrounding this
project and for years ago suggesting to Pierre that Kato estimates for
Schrödinger-type operators would be of particular interest to the community,
thus initiating this line of research.
I am also grateful
to my second supervisor Adam Sikora for his sage
advice, in particular with regards to Proposition <ref>. A
part of this paper was written while visiting him at Macquarie 
University. 

While this paper was in its final stages of preparation,
it was found that Andrew Morris and Andrew Turner from the University
of Birmingham were also working on the Kato problem with potential on ^n. After meeting them and discussing their 
research, it appears that the two approaches differ in their
assumptions, their results
and, more substantially, their proofs.

I would like to thank Moritz Egert for spotting an error in a previous
version of this article and for his comments on Proposition <ref>.
Finally, I would like to thank the anonymous referees of
previous versions of this article for providing such detailed and thoughtful
critiques. Reflecting on the comments provided in their reports led me to several significant improvements.








§ PRELIMINARIES



Let's outline the construction of the natural functional calculus
associated with a bisectorial operator. The treatment of
functional calculi found here follows closely to
<cit.> with significant detail
omitted. Appropriate changes are made to account for the fact that we
consider bisectorial operators instead of sectorial operators. Other
thorough treatments of functional calculus for sectorial operators can
be found in <cit.> or <cit.>.

 For μ∈ [0,π) define the open and closed sectors

    S^o_μ+ := {[ z ∈∖ 0  : arg
                                   z < μ                                           μ∈ (0,π);                                              (0,∞)                                              μ = 0 ].

and

    S_μ+ := {[ z ∈∪∞: arg
                                   z≤μ or  z = 0,  ∞                                                    μ∈ (0,π);                                                    [ 0 , ∞]                                                      μ = 0. ].

Then, for μ∈[0,π/2), define the open and closed bisectors

    S^o_μ := S^o_μ+∪-S^o_μ+

and

    S_μ := S_μ+∪- S_μ +

respectively.  Throughout this section we consider bisectorial operators defined on a
Hilbert space ℋ with norm and inner product denoted by
· and ⟨·, ·⟩ respectively. 

[Bisectorial Operator]
  
   A linear operator T : D T⊆ℋ→ℋ is said
   to be ω-bisectorial for ω∈[0,π/2) if
   the spectrum σT is contained in the bisector
   S_ω and if for any μ∈ω, π/2, there exists C_μ > 0 such that the resolvent
   bound
   
    ζζ I - T^-1≤ C_μ

holds for all ζ∈∖ S_μ. T is said to be
bisectorial if it is ω-bisectorial for some ω∈[0, π/2).


Sectorial operators are defined identically except with the sector
S_μ+ performing the role of the bisector S_μ. An important fact concerning bisectorial
operators is the following decomposition result.


  
Let T : D T⊂ℋ→ℋ be a bisectorial
operator. Then T is necessarily densely defined and
 the Hilbert space ℋ admits the following decomposition
 
    ℋ = N T⊕RT.

 



Let T be an ω-bisectorial operator for ω∈[0,
  π/2) and μ∈ω, π/2. Let ℳS^o_μ denote
the algebra of all meromorphic functions on the open bisector
S^o_μ and define the following subalgebras,

    HS^o_μ :=  f ∈ℳS^o_μ : f holomorphic  on  S^o_μ,


    H^∞S^o_μ :=  f ∈ H S^o_μ :
    f_∞ := sup_z  ∈ S^o_μf(z) < ∞

and

    H^∞_0S^o_μ :=   f ∈ H^∞S^o_μ :
    ∃ C,  α > 0  s.t.  f(z)≤ C ·z^α/1 + z^2 α ∀ z ∈ S^o_μ.

For any f ∈ H^∞_0S^o_μ, one can associate an
operator f(T) as follows. For u ∈ℋ, define

    fTu := 1/2 π i∮_γ f zz I -
      T^-1 u   d z,

where the curve

    γ := ± r e^± i ν : 0 ≤ r < ∞

for some ν∈ (ω, μ) is traversed anticlockwise. Using the
resolvent bounds of our operator in combination with the size
estimates for functions in H^∞_0(S^o_μ), it can
easily be shown that the operator f(T) is well-defined. Moreover, it
can also be proved that this association constitutes an algebra homomorphism.


  
 The map
Φ_0^T : H^∞_0S^o_μ→ℒℋ
defined through

    Φ_0^T(f) := fT

is a well-defined algebra homomorphism. Moreover, it is independent
of the value of ν.


 Since the functions in H^∞_0S^o_μ approach
 zero at the origin we should expect that the null space of the newly
 formed operator will be larger than the null space of the original
 operator. This is indeed the case as stated in the below proposition.


  
 For a bisectorial operator T : D T⊆ℋ→ℋ, the null-space inclusion
 
    N T⊆ N f T

holds for all f ∈ H^∞_0S^o_μ.
 

 Define the subalgebra of functions

    ℰS^o_μ := H^∞_0S^o_μ⊕⟨z + i^-1⟩⊕⟨ 1 ⟩.

Φ_0^T has an extension

    Φ_p^T : ℰS^o_μ→ℒH

defined through 

    Φ_p^T(g) := g(T) := f(T) + c ·T + i^-1 + d · I

for g = f + c ·z + i^-1 + d  ∈ℰS^o_μ, where f ∈ H^∞_0S^o_μ and
c,   d ∈.


  
 The map Φ_p^T is an algebra homomorphism called the primary
 functional calculus associated with T.
 


This map can be extended once more through the  process of
regularization. A function f ∈ℳS^o_μ is said to be regularizable with respect
to the primary functional calculus Φ_p^T : ℰS^o_μ→ℒℋ if there
exists e ∈ℰS^o_μ such that e T is
injective and e · f ∈ℰS^o_μ. The
notation ℰS^o_μ_r will be used to denote the
algebra of regularizable functions. Let 𝒞ℋ
denote the set of closed operators from ℋ to itself. Then define the extension

    Φ^T : ℰS^o_μ_r→𝒞H

through

    Φ^Tf := f(T) := Φ_p^T(e)^-1·Φ_p^T(e · f)

for f ∈ℰS^o_μ_r and e ∈ℰS^o_μ a
regularizing function for f. This definition is independent of the
chosen regularizer e for f and therefore Φ^T is well-defined. We have the following important theorem
that establishes the desired properties of a functional calculus for
this extension. Thus the map Φ^T is known as the natural
functional calculus for the operator T.


  
Let T be an ω-bisectorial operator on a Hilbert space
ℋ for some ω∈[0,π/2). Let
μ∈ω, π/2. The following assertions hold.

  
  * 1T = I and zT = T, where
    1 : S^o_μ→ is the constant function defined by 1(z) :=
    1 for z ∈ S^o_μ.
 
  * Let f,   g ∈ℰS^o_μ_r. Then
   
    f(T) + g(T) ⊂f + g(T),      f(T) g(T) ⊂f · g(T)

and D f(T) g(T) = D
f · g(T)∩ D g(T). One will
have equality in these relations if g(T) ∈ℒ(H).



The ensuing definition plays a vital role in the
solution method to the Kato square root problem using the AKM framework.

 
  
Let 0 ≤ω < μ < π/2. An ω-bisectorial operator
T : D(T) ⊂ℋ→ℋ is said to have
a bounded H^∞S^o_μ-functional calculus if there
exists c > 0 such that 

    f T≤ c ·f_∞

for all f ∈
H^∞_0S^o_μ. T is said to have a bounded holomorphic functional calculus if it
has a bounded H^∞S^o_μ-functional calculus for some μ.



  
Note that a more intuitive definition for a bounded H^∞S^o_μ-functional calculus would be to require that
(<ref>) hold for all f ∈ H^∞S^o_μ. Unfortunately at this stage it is impossible to
ascertain whether
 H^∞S^o_μ⊂ℰS^o_μ_r. When this inclusion does not hold, the
 operator f(T) will not be
 well-defined for all f ∈ H^∞S^o_μ. If T so happens
 to be injective, then each f ∈ H^∞S^o_μ is in fact
 regularizable by z 1 + z^2^-1 and the estimate
 (<ref>) makes sense for all f ∈ H^∞S^o_μ. Fortunately, in this situation the two definitions
 coincide. That is, (<ref>) will be true
 for all f ∈ H^∞_0S^o_μ if and only if it is
 true for all f ∈ H^∞S^o_μ when T is injective. 




[Square Function Norms]
  
 Let ψ∈ H^∞_0S^o_μ.  For t > 0, define ψ_t :
 S^o_μ→ to
 be the function ψ_t(z) := ψ(tz) for z ∈ S^o_μ. The square function
 norm associated with ψ and T is defined through
 
    u_ψ,T := ∫^∞_0ψ_t(T)u^2dt/t^1/2

for u ∈ℋ. Let q : S^o_μ→ be the
function given by

    q(z) := z/1 + z^2,    z ∈ S^o_μ.

·_q,T is called the canonical square function norm
for the operator T.




For injective T, true to its name, the square function norms
·_ψ,T, for ψ∈ H^∞_0S^o_μ  not identically
  equal to zero on either S^o_μ+ or -S^o_μ+,
are indeed norms on ℋ. For non-injective T, however, they
are at most seminorms on ℋ and will only be norms
following a restriction to the subspace R(T).


[Square Function Estimates]
  
  A bisectorial operator T on a Hilbert space ℋ is said
  to satisfy square function estimates if there exists a constant C_SF > 0 such that

    C^-1/2_SF·u≤u_q,T≤ C_SF^1/2·u

for all u ∈R T.


The above definition is the same as saying that the canonical square
function norm ·_q,T is norm equivalent to ·_ℋ when restricted to
the Hilbert subspace R T.



  
  The use of the canonical norm ·_q,T in the above
  definition of square function estimates is somewhat
  arbitrary. Indeed, it can be swapped with ·_ψ,T
  for any ψ∈ H^∞_0S^o_μ not identically
  equal to zero on either S^o_μ+ or -S^o_μ+. This
  follows from the equivalence of these two norms as stated in
  <cit.>. The implicit constant in the
  norm equivalence will depend on the function ψ under consideration.
  




  
 For ψ∈ H^∞_0S^o_μ and any u ∈ℋ,
 
    c_ψI - ℙ_N(T)u = ∫^∞_0ψ_t(T)^2 u  dt/t,

where ℙ_N(T) denotes the projection operator onto the
subspace N T and

    c_ψ := ∫^∞_0ψ(t)^2 dt/t.



  
 Equality follows from Proposition <ref> for
 u ∈ N T. For u ∈R T this is given by
 Theorem 5.2.6 of <cit.> in the sectorial
 case. The bisectorial case can be proved similarly.



  
Suppose that T is self-adjoint and ψ∈ H^∞_0S^o_μ. Then for any u ∈ℋ,
  
    ∫^∞_0ψ_t(T) u^2dt/t≤ c_ψu^2

where c_ψ is as defined in the previous proposition.
Equality will hold if u ∈R T.



  As T is self-adjoint, if follows from the definition of ψ_t(T)
  that it must also be self-adjoint. On expanding the square function norm,
  
    ∫^∞_0ψ_t(T)u^2dt/t   =
     ∫^∞_0⟨ψ_t(T) u, ψ_t(T) u ⟩dt/t
       = ⟨ u , ∫^∞_0ψ_t(T)^2 u dt/t⟩.

The previous proposition then gives

    ∫^∞_0ψ_t(T)u^2dt/t   = ⟨
       u, c_ψI - ℙ_N(T) u ⟩
       ≤ c_ψu^2.

Equality will clearly hold in the above if u ∈R(T).
 




  
Let T be an ω-bisectorial operator on ℋ for
ω∈[0,π/2).  Suppose that T satisfies square function
  estimates with constant C_SF > 0. Then T must have a bounded 
H^∞S^o_μ-holomorphic functional calculus for any
μ∈ω, π/2. In particular, there exists a constant c > 0,
independent of T, such that

    f(T)≤c · C_SF· C_μ·f_∞

for all f ∈  H^∞_0S^o_μ, where C_μ > 0
is the constant from the resolvent estimate (<ref>).




  Let f ∈ H^∞_0S^o_μ.  For u ∈ N T,
  the bound
  
    f T u≤c · C_SF· C_μ·f_∞·u

  follows trivially from Proposition
  <ref> for any c > 0. Fix u ∈R
    T. On applying the lower square function estimate to f(T) u
  ∈R T,
  
    f(T) u^2   ≤ C_SF∫_0^∞q_s(T) fT u^2ds/s
       = 2 C_SF∫^∞_0q_s(T) fT∫_0^∞q_t(T)^2 u dt/t^2ds/s
       ≤ 2 C_SF∫^∞_0∫^∞_0q_s(T) fT q_t(T)q_t(T) udt/t^2ds/s,

where in the second line we used the resolution of the identity
Proposition <ref>.
The Cauchy-Schwarz inequality then gives

    f T u^2   ≤ 2 C_SF∫^∞_0∫^∞_0q_s(T) f T q_t(T)dt/t∫^∞_0q_s(T) f T
       q_t(T)q_t(T) u^2dt/tds/s.

From the homomorphism property for the H^∞_0S^o_μ-functional calculus,

    q_s(T) f
    T q_t(T) = q_s· f · q_tT.

 Since our operator T satisfies resolvent bounds with constant
 C_μ > 0,

    q_s(T) f(T) q_t(T)   = q_s· f · q_tT
       = 1/2 π∫_γq_s· f · q_t(z)
       T - z I^-1 dz
       ≤C_μ/2 π·f_∞·∫_γq_s(z)q_t(z)dz/z.

On noting that q ∈ H^∞_0S^o_μ,

    q_s(T) f T q_t(T)≤
    c · C_μ·f_∞·∫_γs
         z^α/1 + s z^2 αt
         z^α/1 + t z^2 αdz/z

 for some α > 0 and constant c > 0 independent of T. Thus we obtain the estimate
 
    q_s(T) f T q_t(T)≤
    c · C_μ·f_∞·{[                                                                                                                                                                                                                                                                                                                                                   t/s^α1 + logs/t                                                                                                                                                                                                                        for  0 <
                                                                t ≤ s <
                                                                ∞;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    ;                                                                                                                                                                                                                                                                                                                                                   s/t^α1 + logt/s for
                                                                                                                                          0 <
                                                                       s
                                                                       <
                                                                       t < ∞, ].

where the value of the T independent constant c is allowed to change. This then implies that

    sup_s > 0∫^∞_0q_s(T)f(T)
      q_t(T)dt/t,   sup_t > 0∫^∞_0q_s(T) f(T) q_t(T)ds/s≤ c · C_μ·f_∞.

On applying this estimate to (<ref>), 

    f T u^2   ≤ c · C_μ· C_SF·f_∞∫^∞_0∫^∞_0q_s(T) f
      T q_t(T)q_t(T) u^2dt/tds/s
       = c · C_μ· C_SF·f_∞∫^∞_0q_t(T) u^2∫^∞_0q_s(T) f
      T q_t(T)ds/sdt/t
       ≤ c^2· C_μ^2· C_SF·f_∞^2∫_0^∞q_t(T) u^2dt/t
       ≲ c^2· C_μ^2· C_SF^2·f_∞^2u^2.

 


Finally, the following Kato-type estimate follows from a well-known
classical argument.



  
  Suppose that the bisectorial operator T satisfies square function
  estimates with constant C_SF > 0 and the constant in the resolvent
  estimate (<ref>) is C_μ > 0. Then there
  exists a constant c > 0, independent of T, such that
  
    c · C_SF· C_μ^-1·T u≤√(T^2)u≤c · C_SF· C_μ·T u

 for any u ∈ DT.


  
Consider the restriction S := T |_RT. S is
an injective bisectorial operator that satisfies square function
estimates with constant C_SF > 0. Since S is injective it follows that f(S) is
well-defined for any f ∈ H^∞S^o_μ by Remark
<ref>. This allows us to define the operators f_1(S)
and f_2(S), where the functions f_1 and f_2 are defined by

    f_1(z) := √(z^2)/z   and    f_2(z) :=
    z/√(z^2)   for    z ∈ S^o_μ.

The previous theorem allows us to deduce that both of these operators
are norm bounded by c · C_SF· C_μ for some T
independent constant c > 0. Applying the multiplicative part of Theorem
<ref> to the functions f = f_1 and g(z) = z
implies that

    √(S^2)/S· S = √(S^2)

on

    D(S) = D √(S^2)/S· S = D √(S^2)∩ D S.

Similarly, applying the multiplicative part of Theorem
<ref> to f = f_2 and g(z) = √(z^2) gives

    S/√(S^2)·√(S^2) = S

on

    D √(S^2) = D S/√(S^2)·√(S^2) = D S∩ D √(S^2).

Equations (<ref>) and (<ref>) together
imply that the domains D √(S^2) and D S coincide
and therefore both (<ref>) and (<ref>)
will remain valid on all of D S.



Let u ∈ D T. Proposition <ref> states that u has the decomposition u = u_1⊕
u_2∈ N T⊕R T. Then

    T u   = S u_2
       = S/√(S^2)·√(S^2) u_2
       ≤ c · C_SF· C_μ·√(S^2) u_2
       = c · C_SF· C_μ·√(T^2)u,

where in the last line we used the fact that the functional
calculus commutes with the restriction map as given in Proposition
2.6.5 of <cit.>. Also,

    √(T^2)u   = √(S^2)u_2
       = √(S^2)/S· S u_2
       ≤ c · C_SF· C_μ·S u_2
       = c · C_SF· C_μ·T u.








 




§ NON-HOMOGENEOUS AXELSSON-KEITH-MCINTOSH



In this section we describe how the Axelsson-Keith-McIntosh framework
can be altered to account for non-homogeneous operators of the form (<ref>). Our  main
results for this framework will also be stated.




 §.§ AKM without Cancellation and Coercivity



The operators that we wish to consider, Γ_J,
will satisfy the first six conditions of
<cit.>. However, they will not necessarily
satisfy the cancellation condition (H7) and the coercivity condition
(H8). It will therefore be fruitful to see what happens to the
original AKM framework when the cancellation and coercivity conditions
are removed.

Similar to the original result, we begin by assuming that we have operators that satisfy the
hypotheses (H1) - (H3) from <cit.>. Recall these
conditions for operators Γ, B_1 and B_2 on a Hilbert
space ℋ with norm · and inner product
⟨·, ·⟩.







  (H1) Γ : D(Γ) →ℋ is a closed,
  densely defined, nilpotent operator.
  


  (H2) B_1 and B_2 satisfy the accretivity conditions
  
    Re⟨ B_1 u, u ⟩≥κ_1u^2     and     Re⟨ B_2 v, v ⟩≥κ_2v^2

for all u ∈ R(Γ^*) and v ∈ R Γ for some
κ_1, κ_2 > 0.



  (H3) The operators Γ and Γ^* satisfy
  
    Γ^* B_2 B_1Γ^* = 0      and     Γ B_1 B_2Γ
    = 0.
 



In <cit.> Section 4, the authors assume that they
have operators that satisfy the
hypotheses (H1) - (H3) and they derive several important operator
theoretic consequences from only these hypotheses. As our operators
Γ, B_1 and B_2 also satisfy (H1) - (H3), it follows
that any result proved in <cit.>
must also be true for our operators and can be used with
impunity.  In the
interest of making this article as self-contained as possible, we will now
restate any such result that is to be used in this paper.



  
 Suppose that the operators Γ, B_1, B_2 satisfy (H1)
- (H3). Define the perturbation dependent operators 

    Γ^*_B := B_1Γ^* B_2,   Γ_B :=
    B_2^*Γ B_1^*   and   Π_B := Γ + Γ_B^*.

The Hilbert space ℋ has the following Hodge decomposition
 into closed subspaces:

    ℋ = N Π_B⊕R Γ^*_B⊕R Γ.

Moreover, we have N Π_B = N Γ^*_B∩ N
Γ and R Π_B = R
  Γ^*_B⊕R Γ. When B_1 =
B_2 = I these decompositions are orthogonal, and in general the
decompositions are topological. Similarly, there is also a
decomposition

    ℋ = N Π_B^*⊕R Γ_B⊕R Γ^*.

 


  
  Suppose that the operators Γ, B_1, B_2 satisfy (H1)
- (H3).
  The perturbed Dirac-type operator Π_B is an ω-bisectorial operator with ω
  := 1/2ω_1 + ω_2 where
  
    ω_1 := sup_u ∈ R Γ^*∖ 0 arg⟨ B_1 u, u ⟩ < π/2

and

    ω_2 := sup_u ∈ R Γ∖ 0 arg⟨ B_2u, u ⟩ < π/2.

 

 The bisectoriality of Π_B ensures that the following operators
 will be well-defined.

 
   
Suppose that the operators Γ, B_1, B_2 satisfy (H1)
- (H3). For t ∈∖ 0, define the perturbation dependent operators

    R^B_t := I + i t Π_B^-1,    P^B_t := I +
      t^2Π_B^2^-1,


    Q_t^B := t Π_B P_t^B   and   Θ_t^B := t Γ^*_B P_t^B.

When there is no perturbation, i.e. when B_1 =
B_2 = I, the B will dropped from the superscript or subscript. For example,
instead of Θ^I_t or Π_I the notation Θ_t and
Π will be employed.



An easy consequence of Proposition <ref> is that
the operators R^B_t, P^B_t and Q^B_t are all uniformly
ℋ-bounded in t. Furthermore, on taking the Hodge decomposition
Proposition <ref> into account, it is clear that the
operators Θ^B_t will also be uniformly ℋ-bounded in
t.



The next result tells us how the operators Π_B and P^B_t interact
with Γ and Γ^*_B.


  
  Suppose that the operators Γ, B_1, B_2 satisfy (H1)
- (H3). The following relations are true.
 
    Π_BΓ u = Γ^*_BΠ_B u    for  all  u ∈ D
     Γ^*_BΠ_B,

 
    Π_BΓ^*_B u = ΓΠ_B u    for  all  u ∈ D
     ΓΠ_B,

 
    Γ P_t^B u = P^B_tΓ u    for  all  u ∈ D
    Γ,    and


    Γ^*_B P^B_t u = P^B_tΓ^*_Bu     for  all
     u ∈ D Γ^*_B.

 

The subsequent lemma provides a square function estimate for the
unperturbed Dirac-type operator Π. When considering square
function estimates for the perturbed operator, there will be several
instances where the perturbed case can be reduced with the assistance
of this unperturbed estimate. Its proof follows directly from the
self-adjointness of the operator Π and Corollary <ref>.



  
 Suppose that the operators Γ, B_1, B_2 satisfy (H1)
- (H3). The quadratic estimate

    ∫^∞_0Q_tu^2dt/t≤1/2u^2

holds for all u ∈ℋ. Equality holds on RΠ.
 

The following result will play a crucial role in
the reduction of the square function estimate (<ref>). 

 


  
 Suppose that the operators Γ, B_1, B_2 satisfy (H1)
- (H3). Assume that the estimate
 
    ∫^∞_0Θ^B_t P_t u^2dt/t≤ c ·u^2

 holds for all u ∈ R Γ and some constant c > 0, together
 with three similar estimates obtained on replacing Γ,
 B_1, B_2 by Γ^*, B_2, B_1, Γ^*, B_2^*, B_1^* and Γ, B_1^*,
 B_2^*. Then Π_B satisfies the quadratic estimate
 
    c · C^-1·u^2≤∫^∞_0Q^B_t u^2dt/t≤ c · C ·u^2

for all u ∈R Π_B, for some C > 0 entirely
dependent on (H1) - (H3).


The constant dependence of (<ref>) is not
explicitly mentioned in Proposition 4.8 of <cit.>, but it is
relatively easy to trace through their argument and record where
(<ref>) is used. The following corollary is proved
during the course of the proof  of 
<cit.>. 


  
  Suppose that the operators Γ, B_1, B_2 satisfy (H1)
- (H3). There exists a constant c > 0 such
  that for any u ∈ RΓ,
  
    ∫^∞_0Θ^B_tI - P_tu^2dt/t≤ c ·u^2.

  



From this point onwards, it will also be assumed that our operators
satisfy the additional hypotheses (H4) - (H6). These hypotheses are
stated below for reference. 







  (H4) The Hilbert space is ℋ = L^2^n;
    ^N for some n,   N ∈^*.
  

  
  (H5) The operators B_1 and B_2 represent
    multiplication by matrix-valued functions. That is,
    
    B_1(f)(x) = B_1(x) · f(x)      and      B_2(f)(x) =
    B_2(x) · f(x)

for all f ∈ℋ and x ∈^n, where B_1, B_2∈ L^∞^n; ℒ^N.


 
  (H6) For every bounded Lipschitz function η : ^n→, we have that η D(Γ) ⊂ D
   Γ and η D Γ^*⊂ D
   Γ^*. Moreover, the commutators Γ,
     η I and Γ^*, η I are 
   multiplication operators that satisfy the bound
   
    Γ, η I(x),  Γ^*, η I(x)≤ c ∇η(x)

for all x ∈^n and some constant c > 0.








In contrast to the original result, our operators will not be assumed to satisfy the cancellation
condition (H7) or the coercivity condition (H8). Without these two
conditions, many of the results from Section 5 of
<cit.> will fail. One notable exception to this
is that the bounded operators associated with our perturbed
Dirac-type operator Π_B will satisfy off-diagonal estimates.

[Off-Diagonal Bounds]
  
  Define
  ⟨ x ⟩ := 1 + x
  for x ∈ and dist(E,F) := infx - y : x
  ∈ E, y ∈ F for E, F ⊂^n.
  
Let U_t_t > 0 be a family of operators on ℋ =
L^2^n;^N.
 This collection is said to have off-diagonal bounds of order M
 > 0 if there exists C_M > 0  such that
 
    U_tu_L^2E≤ C_M⟨dist(E,F) / t
      ⟩^-Mu

  whenever E,   F ⊂^n  are Borel sets and u ∈ℋ satisfies supp  u ⊂ F. 




 
   
   Suppose that the operators Γ, B_1, B_2 satisfy
   (H1) - (H6).
 Let U_t be given by either R^B_t,
R^B_-t, P^B_t, Q^B_t or Θ^B_t for every t >
0. The collection of operators U_t_t
  > 0  has off-diagonal bounds of every order M > 0. Moreover, the constant C_M in the estimate
(<ref>) depends only on M and the
hypotheses (H1) - (H6).


Introduce the following dyadic decomposition of ^n. Let Δ
= ∪_j = - ∞^∞Δ_2^j where Δ_t :=
 2^j(k + (0,1]^n) : k ∈ℤ^n if 2^j-1 < t
  ≤ 2^j for j ∈ℤ. Define the averaging operator A_t : ℋ→ℋ through

    A_tu(x) := _Q(x,t) u(y)   dy := 1/Q(x,t)∫_Q(x,t) u(y)   dy

for x ∈^n, t > 0 and u ∈ℋ, where Q(x,t) is
the unique dyadic cube in Δ_t that contains the point x.


For an operator family U_t_t > 0 that satisfies
off-diagonal bounds of every order, there exists an extension U_t:
L^∞^n; ^N→ L^2_loc^n;^N for each t > 0. This is constructed by defining 

    U_t u(x) := lim_r →∞∑_R ∈Δ_t
    distQ,R < r U_t1_R u(x),

for x ∈ Q ∈Δ_t and u ∈ L^∞^n;
  ^N. The convergence of the above limit is guaranteed
by the off-diagonal bounds of U_t_t > 0. Further detail
on this construction can be found in <cit.>,
<cit.>, <cit.> or <cit.>. The above
extension then allows us to introduce the principal part of the
operator U_t.




 
  
 Let U_t_t > 0 be operators on ℋ that satisfy
 off-diagonal bounds of every order. For t > 0, the principal part of U_t is
 the operator ζ_t : ^n→ℒ^N
 defined through
 
    ζ_t(x)w := U_t w(x)

for each x ∈^n and w ∈^N.
 

The following generalisation of Corollary 5.3 of
<cit.> will also be true with an identical proof.


  
Let U_t_t > 0 be operators on ℋ that satisfy 
off-diagonal bounds of every order. Let ζ_t : ^n→ℒ^N denote the principal part of the operator
U_t. Then there exists c > 0 such that

    _Qζ_t(y)^2 dy ≤ c

for all Q ∈Δ_t and t > 0. Moreover, the operators ζ_t A_t
are uniformly L^2-bounded in t.


Finally, the ensuing partial result will also be valid. Its proof follows in
an identical manner to the first part of the proof of Proposition 5.5
of <cit.>.


  
  
 Let U_t_t > 0 be operators on ℋ that satisfy
 off-diagonal bounds of every order. Let ζ_t : ^n→ℒ^N denote the principal part of U_t. Then
 there exists c > 0 such that
 
    U_t - ζ_t A_t v≤ c ·t ∇ v.

 for any v ∈ H^1^n;^N⊂ℋ and t > 0.
 

 




 

 §.§ Additional Structure

 
 
 
At this point, further structure will be imposed upon  our
operators in order to generalise the non-homogeneous operator Γ_V^1/2 defined in
(<ref>). This additional structure will later be exploited in order to obtain square
function estimates of the form (<ref>).




Let ^N = V_1⊕ V_2⊕ V_3
where V_1,   V_2 and V_3 are finite-dimensional complex Hilbert spaces. Let ℙ_i : ^N→^N be
the projection operator onto the space V_i for i = 1,   2 and 3. Our Hilbert space will have the following
orthogonal decomposition

    ℋ := L^2^n; ^N =  L^2^n;V_1⊕ L^2^n;V_2⊕ L^2(^n;V_3).

The notation ℙ_i will also be used to denote the natural
projection operator from ℋ onto
L^2^n;V_i. For a vector v ∈ℋ, v_i∈
L^2^n;V_i will denote the ith component for i = 1,  
2 or 3.

Let Γ_J be an operator on ℋ of the form

    Γ_J := [   0   0; D_J   0 ] := [ 0 0 0; J 0 0; D 0 0 ],

where J and D are closed densely defined operators

    J : L^2^n;V_1→ L^2^n;V_2 and


    D : L^2^n;V_1→ L^2^n;V_3,

and D_J : L^2^n;V_1→ L^2^n;V_2⊕ L^2^n;V_3 is the operator D_J = [ J; D ]. Define the operators

    Γ_0 := [ 0 0 0; 0 0 0; D 0 0 ],      M_J := [ 0 0 0; J 0 0; 0 0 0 ],




    Π_0 := Γ_0 + Γ_0^*,     S_J := M_J +
    M_J^*   and   Π_J := Γ_J + Γ_J^*.





Let B_1,   B_2∈ L^∞^n; ℒ^N be matrix-valued multiplication operators. The
following key assumption will be imposed on our operators throughout
the entirety of this article.




33em
Key Assumption. The family of operators
Γ_0, B_1, B_2 satisfies the conditions (H1) -
(H8) of <cit.> while Γ_J, B_1,
B_2 satisfies (H1) - (H6).






For reference, the cancellation condition (H7)
and the coercivity condition (H8) are shown below for the operator
Γ_0.






  
  (H7) For any u ∈ D Γ_0 and v ∈ D
    Γ^*_0, both compactly supported,
    
    ∫_^nΓ_0 u = 0    and   ∫_^nΓ^*_0 v
    = 0.






  (H8)  There  exists c > 0 such that
  
    ∇ u≤ c ·Π_0 u

for all u ∈ R Π_0∩ D Π_0.
  

  


  
    Typical examples of operators that satisfy the previous key
    assumption are when both D and J are partial differential
    operators of order less than or equal to one. If 
the perturbations B_1 and B_2 satisfy suitable accretivity
conditions then the families of operators Γ_0, B_1, B_2 and Γ_J, B_1, B_2 will both satisfy (H1) - (H6). If, in
addition, D is first-order homogeneous and there exists c > 0 for which

    ∇ u≤ c ·D u

for all u ∈ R D^*∩ D D and

    ∇ v≤ c ·D^*v

for all v ∈ R D∩ D D^* then Γ_0, B_1,
B_2 will also satisfy (H7) and (H8). A particular example of
such a situation is given by the operator
Γ_V^1/2 together with perturbations B_1
and B_2 as defined in (<ref>) and
(<ref>) with (<ref>) and
(<ref>) satisfied.
    





Since the operator Γ_0, together with the
perturbations B_1 and B_2, satisfy  all
eight conditions (H1) - (H8) of <cit.>, it follows that any
result from that paper must be valid for these operators.




 
  
For t ∈∖ 0, define the perturbation dependent operators

    Γ_J,B := B_2^*Γ_J B_1^*,   Γ_J,B^* := B_1Γ^*_J B_2,   Π_J,B :=
    Γ_J + Γ_J,B^*,


    R^J,B_t := I + i t Π_J,B^-1,    P^J,B_t := I +
      t^2Π_J,B^2^-1,


    Q_t^J,B := t Π_J,B P_t^J,B   and   Θ_t^J,B := t Γ^*_J,B P_t^J,B.

When there is no perturbation, i.e. when B_1 =
B_2 = I, the B will dropped from the superscript or subscript. For example,
instead of Θ^J,I_t the notation Θ^J_t will be
employed.




 We now introduce coercivity conditions to serve as a
replacement for (H8) for the operators Γ_J, B_1, B_2.  These conditions will not be automatically imposed
upon our operators but, rather, will be taken as hypotheses for
our main results.






  (H8Dα) Let α∈ (1,2]. The domain inclusion
  
    D(D_J^*D_J)^α/2⊂ D (D^*D)^α/2

holds and there exists a
constant C > 0 such that

    D^* D^α/2 u≤ C ·D_J^*D_J^α/2u

for all u ∈ D (D_J^*D_J)^α/2.







  (H8J) B_2 is of the form
  
    B_2 = [ I 0; 0 Â ] := [    I    0    0;    0 A_22 A_23;    0 A_32 A_33 ],

where A_ij∈ L^∞^n; ℒV_j,
    V_i for i,   j = 2 or 3. The inclusion
  
    D D_J^* D_J⊂ D D_J^*Â[ J; 0 ]

is satisfied. Furthermore, there exists a constant C > 0 such that for all u
  ∈ D D_J^* D_J,
  
    D_J^*Â[ J; 0 ]u≤ C ·D_J^* D_J u.








  
  The situation of most interest to us is when A_32 = 0 and
  
    J^* A_22 J u = J^*J u

for all u ∈ D J^* A_22 J = D J^* J. In this case,
the domain inclusion of (H8J) becomes D(D^*_J D_J) ⊂
D(J^*J) and the Riesz transform condition becomes the perturbation free condition

    J^*J u≤ C ·D_J^* D_J u

for all u ∈ D D_J^* D_J. Furthermore, when this occurs, the Riesz
transform condition of (H8J) will
be equivalent to the condition

    S_J u≤ C ·Π_J u

or equivalently

    Π_0 u≤ C ·Π_J u

for all u ∈ D Π_J∩ R Π_J. 
  

  


  
The Kato square root estimate is a first-order Riesz transform
condition. To some extent, it then seems intuitively unnatural to use a
second-order Riesz transform condition as in (H8J) as our
hypotheses. Indeed, when the conditions of the above remark are satisfied and J is a
  positive operator, it will be
  sufficient to consider a lower-order version of (H8J) as given below.

  




  (H8Jα) Let α∈ (1,2]. The perturbation B_2
  is of the form (<ref>) with A_32 = 0. J is a
  positive operator and
  
    J A_22 J u = J^2 u

for all u ∈ D J A_22 J = D J^2. The domain inclusion

    D (D_J^* D_J)^α/2⊂ D J^α

holds. Furthermore, there exists a
constant C > 0 such that

    J^α u≤ C ·D_J^* D_J^α/2u

for all u ∈ D (D_J^* D_J)^α/2.






  For α∈ (1,2], let b^D_α,   b^J and
  b^J_α denote the smallest constant for which
  (H8Dα), (H8J) or (H8Jα) are satisfied respectively. If the criteria
  for one of these conditions is not met then the corresponding
  constant will be set to infinity. For example, if (H8J) is not
  satisfied then b^J = ∞. Also define
  
    c_α^J := 1 + b^D_α^2 + min
        b^J, b^J_α^2· (α - 1)^-1.

For the remainder of this article we
introduce the notation A ≲ B and A ≃ B to denote that there exists a constant C >
0, independent of (H8Dα), (H8J) and (H8Jα), for which A ≤ C · B and C^-1· B ≤ A ≤ C · B respectively.  C is still
allowed to depend on (H1) - (H8) for Γ_0, B_1, B_2 and (H1) - (H6) for Γ_J, B_1, B_2. Note that
this implies that C will only depend on J through the constants in
(H2) for Γ_J, B_1, B_2 and it will be independent
of α.


We are now in a position
where the main result of the non-homogeneous AKM framework can finally
be stated.

 
  
Let Γ_J, B_1, B_2 be as defined
above. Consider the following square function estimate:
 
    ∫^∞_0Θ^J,B_tℙ_i P_t^Ju^2dt/t≲ C ·u^2

for all u ∈ R Γ_J, for some C > 0,  for i = 1,   2 or
3.





  
  *  The estimate is trivially satisfied for i = 1 for any
    C ≥ 0.





  *  If (H8J) is satisfied then (<ref>) is satisfied
  for i = 2 with C = b^J^2.



  
  
  *  If (H8Jα) is satisfied for some α∈ (1,2]
    then (<ref>) is satisfied
for i = 2 with C = (1+b_α^J^2) (α - 1)^-1.





  *  If (H8Dα) is satisfied for some α∈ (1,2] and
  either (H8J) or (H8Jα) is also satisfied then
(<ref>) holds for i = 3 with constant C = c_α^J.







  
  The proof of (iii) and (iv) will be postponed until Section
  <ref>. For (i), simply note that since Γ_J
  commutes with the operator P_t^J by Lemma <ref> we must have ℙ_1
  P_t^J u = 0 for any u ∈ R Γ_J.
  
It remains to  consider (ii).   Suppose that (H8J) holds. First it will be proved that for u ∈ R
  Γ_J we have ℙ_2 P_t^J u ∈ D
  Γ_J,B^*.
 Since u ∈ R Γ_J, u = Γ_J v for some
v ∈ D Γ_J. As

    P_t^J u = P_t^JΓ_J v = Γ_J P_t^J v

by Lemma <ref> and P_t^J u ∈ D Π_J,
it follows that P_t^J v_1∈ D D_J^* D_J,
which by (H8J) is contained in D D_J^*Â[ J; 0 ]. This implies that

    [ J (P_t^J v)_1;             0 ] = [ (P_t^Ju)_2;          0 ]∈ D (D_J^*Â)

and therefore ℙ_2 P_t^J u ∈ D (Γ^*_J,B).

Since ℙ_2 P_t^J u ∈ D Γ_J,B^*, it
follows from Lemma <ref> that

    Θ^J,B_tℙ_2 P_t^Ju = P^J,B_t
    t Γ^*_J,Bℙ_2 P_t^Ju.

The estimate in (H8J) gives 

    Γ_J,B^*ℙ_2ṽ≤ b^J·Γ^*_Jṽ

for any ṽ∈ R Γ_J∩ D (Π_J). Since P_t^J and Γ_J
commute by Lemma <ref>, it follows that

    Γ_J,B^*ℙ_2 P_t^J u≤ b^J·Γ^*_J P_t^J u
 
for u ∈ RΓ_J.  On applying the uniform
L^2-boundedness of the P_t^J,B operators,

    ∫^∞_0Θ^J,B_tℙ_2 P_t^J
          u^2dt/t   =
     ∫^∞_0P_t^J,B t Γ^*_J,Bℙ_2 P_t^J u^2dt/t
       ≲∫^∞_0t Γ_J,B^*ℙ_2 P_t^J
       u^2dt/t
       ≤b^J^2∫^∞_0t Γ_J^* P_t^Ju^2dt/t.

 On successively applying
Proposition <ref> and Lemma <ref> we obtain

    ∫^∞_0Θ^J,B_tℙ_2 P_t^J u^2dt/t   ≲b^J^2∫^∞_0t Π_J P_t^J
       u^2dt/t
       = b^J^2∫^∞_0Q_t^J u^2dt/t
       = 1/2b^J^2u^2.

This shows that (<ref>) is valid for i = 2 with constant C = b^J^2. 
 







Let's consider an estimate that serves as a dual to (<ref>). 


 
  
For t > 0, define the operator
 
    P^J,B_t := I + t^2Γ_J^* + B_2Γ_J B_1^2^-1.

The square function estimate

    ∫^∞_0 t B_2Γ_J B_1P^J,B_t
       P_t^J u^2dt/t≲u^2

will hold for all u ∈ℋ when B_1 = I.
 

 
   
Since Γ_J, B_1, B_2 satisfies (H1) - (H6) it
follows that Γ_J^*, B_2, B_1 must also satisfy
(H1) - (H6). Proposition <ref> then implies that the operators
P_t^J,B are well-defined  and uniformly
L^2-bounded. Since B_1 = I, it follows that P_t^J u ∈
D(B_2Γ_J B_1) for any u ∈ℋ and therefore,
by Lemma <ref>,

    B_2Γ_J B_1P_t^J,B P_t^J u =
    P_t^J,B B_2Γ_J B_1 P_t^J u =
    P_t^J B_2Γ_J P_t^J u.

This together with the uniform L^2-boundedness of the
P_t^J,B operators implies that

    ∫^∞_0t B_2Γ_J B_1P_t^J,B P_t^J u^2 dt/t   =   ∫^∞_0P_t^J,B t B_2Γ_J 
       P_t^Ju^2dt/t
       ≲∫^∞_0t Γ_J P_t^J u^2dt/t
       ≤∫^∞_0t Π_J P_t^J u^2dt/t
       = ∫^∞_0Q_t^Ju^2dt/t
       = 1/2u^2,

where the inequality Γ_J v≤Π_J v for v
∈ D Π_J follows immediately from the three-by-three matrix form
of the operators and Lemma <ref> was applied to obtain
the last line.




 
From our main result, Theorem <ref>, and the previous proposition, the upper and lower square function
estimates for Q^J,B_t can be proved. 

 
  
 Suppose that B_1 = I. Suppose further that
(H8Dα) is satisfied for some α∈ (1,2] and either
(H8J) or (H8Jα) is satisfied. Then
 
    c_α^J^-1·u^2≲∫^∞_0Q^J,B_tu^2dt/t≲ c_α^J·u^2

for all u ∈R Π_J.


  
 Proposition <ref> states that in order to prove the square function estimate
 (<ref>), it is sufficient for the estimate
 (<ref>) to be valid for all i = 1,   2 and 3 for the permutations of operators
 Γ_J, B_1, B_2, Γ_J, B_1^*,
 B_2^*, Γ_J^*,B_2, B_1 and Γ_J^*, B_2^*, B_1^*. The permutations Γ_J, B_1, B_2 and Γ_J, B_1^*,
 B_2^* both come under the umbrella of Theorem
 <ref> and the permutations Γ_J^*, B_2,
 B_1 and Γ_J^*, B_2^*, B_1^* are
 handled by Proposition <ref>.



From the upper and lower estimate of the previous theorem, Theorem
<ref> then implies that Π_J,B has a bounded holomorphic functional calculus.

 
  
 Suppose that B_1 = I. Suppose further that
(H8Dα) is satisfied for some α∈ (1,2] and either
(H8J) or (H8Jα) is satisfied.  Then Π_J,B has a bounded
H^∞S^o_μ-holomorphic functional calculus for any
μ∈ω_J, π/2, where

    ω_J := 1/2sup_u ∈ R Γ_J^*∖ 0 arg⟨ B_1 u, u ⟩
    + sup_u ∈ R Γ_J∖ 0 arg⟨
      B_2 u, u
    ⟩.

 In particular,

    f(Π_J,B)≲ c_α^J·sup_ζ∈
      S^o_μf(ζ)

for any f ∈ H^∞_0S^o_μ. 




  
 Suppose that B_1 = I. Suppose further that
(H8Dα) is satisfied for some α∈ (1,2] and either
(H8J) or (H8Jα) is satisfied. The operator

    L_B^J := D_J^*Â D_J

is a 2 ω_J-sectorial operator with  a bounded
H^∞S^o_2μ+-functional calculus for any μ∈ω_J, π/2 and

    √(L_B^J) u≃ c_α^J·J u +
      D u

for all u ∈ D J∩ D(D).
  

  
The bounded H^∞S^o_2 μ+-functional calculus of
L_B^J follows from the bounded
H^∞S^o_μ-functional calculus of Π_J,B and
that Π_J,B^2 is of the form

    Π_J,B^2 = [ L_B^J     0     0;     0     *     *;     0     *     * ].

The estimate (<ref>) follows from Corollary
<ref> applied to the operator Π_J,B and an element
(u,0,0) ∈ℋ with u ∈ D J∩ D(D).



  






§ SQUARE FUNCTION ESTIMATES



In this section, a proof of our main result, Theorem
<ref>, will be provided. The first part of the proof
consists in showing that the operators P_t^J can effectively be
diagonalised when estimating square function norms from above. This diagonalisation
will be applied to bound the second component of our square
function norm when (H8Jα) is satisfied thus proving the third
part of Theorem <ref>.
To prove the fourth and most challenging part of Theorem <ref> we will use this diagonalisation and an argument similar to
the original result <cit.>. That is, a T(1)-type
reduction will be applied to reduce the third component of the square
function norm to a Carleson measure norm which will subsequently be
proved to be bounded.




 §.§ Diagonalisation of the P_t^J Operators


Define the bounded operator 𝒫_t^J : ℋ→ℋ through

    𝒫_t^J := [ I + t^2 D_J^* D_J^-1                    0                    0;                    0     I + t^2 J J^*^-1                    0;                    0                    0     I + t^2 D D^*^-1 ],

for t > 0. Observe that since the operators D_J^* D_J,
J J^* and D D^* are all self-adjoint, it follows from Corollary
<ref> that square function estimates hold for each of these
operators with constant independent of J and D. Therefore each of
these operators possesses a bounded holomorphic functional calculus
with constant independent of J and D. It can be deduced from this
that the operators 𝒫_t^J are uniformly L^2-bounded
with constant independent of J and D.

Let us prove that the operator P_t^J can be
effectively diagonalised when evaluating square function
estimates. Specifically, the following theorem will be proved.

          
 
  
 Suppose that (H8Dα) is satisfied for some α∈ (1,2]
 then
 
    ∫^∞_0ℙ_3(𝒫_t^J -
      P_t^J)u^2dt/t≲ (1 + (b^D_α)^2)
    ·u^2

for all u ∈ R(Γ_J). Suppose, in addition, that
(H8Jα) is also satisfied. Then
 
    ∫^∞_0𝒫_t^J -
        P_t^Ju^2dt/t≲ c_α^J·u^2

for all u ∈ R Γ_J.



 Such a
diagonalisation will aid us tremendously in the bounding of our main
square function estimate (<ref>) for the second and
third component.  This theorem will be proved by inspecting each component separately.



  
It is easy to see that the diagonalisation estimate
(<ref>) is trivially satisfied on the first component
for any u ∈ℋ since
ℙ_1𝒫_t^J = ℙ_1 P_t^J. 



   
  
 For any u ∈ℋ,
 
    ∫^∞_0𝒫_t^JP_t^J - I u^2dt/t≲u^2.





    The estimate is trivially satisfied for any u ∈ N(Π_J) since

    P_t^J - I u    = I + t^2Π^2_J^-1 - I u 
       = I + t^2Π^2_J^-1I - I + t^2Π^2_J u 
       = 0

for any t > 0. Suppose that
u ∈R Π_J. On applying the resolution of the
identity, Proposition <ref>,

    ∫^∞_0𝒫_t^JP_t^J - I u^2dt/t   = ∫^∞_0𝒫_t^JP_t^J - I
      2 ∫^∞_0Q_s^J^2 u ds/s^2dt/t
       ≲∫^∞_0∫^∞_0𝒫_t^JP_t^J - IQ_s^J^2 uds/s^2dt/t.

The Cauchy-Schwarz inequality leads to

    ∫^∞_0   𝒫_t^JP_t^J - I u^2dt/t≲
         ∫^∞_0∫^∞_0𝒫_t^JP_t^J - I Q_s^Jds/s·∫^∞_0𝒫_t^JP_t^J - I Q_s^JQ_s^Ju^2ds/sdt/t.

Let's estimate the term 𝒫_t^JP_t^J - I Q_s^J. First
assume that t ≤ s. On noting that P_t^J - I Q_s^J =
t/s Q_t^JP_s^J - I we obtain

    𝒫_t^JP_t^J - I Q_s^J≲P_t^J - I
       Q_s^J≲t/sQ_t^JP_s^J - I≲t/s.

Next, suppose that t > s. Then the equality P_t^J Q_s^J =
s/t Q_t^J P_s^J gives

    𝒫_t^JP_t^J - IQ_s^J≲P_t^J Q_s^J +
    𝒫_t^J Q_s^J≲s/t
    + 𝒫_t^J Q_s^J.

The term 𝒫_t^J Q_s^J will be considered
component-wise. For the first component, recall that ℙ_1𝒫_t^J = ℙ_1 P_t^J and observe that

    ℙ_1𝒫_t^J Q_s^J   =
     ℙ_1 P_t^J s Π_J P_s^J
       = s/tℙ_1 P_t^J t Π_J P_s^J
       = s/tℙ_1 Q_t^J P_s^J
       ≲s/t.

For the second component, note that

    ℙ_2𝒫_t^J = I + t^2 S_J^2^-1ℙ_2.

Also observe

    ℙ_2Π_J u = ℙ_2Π_Jℙ_1 u =
    ℙ_2 S_Jℙ_1 u

for u ∈ D(Π_J).
This gives

    ℙ_2𝒫_t^J Q_s^J   =
     I + t^2 S_J^2^-1ℙ_2 s Π_J
       P_s^J
       = (I + t^2 S_J^2)^-1ℙ_2 s S_Jℙ_1 P_s^J
       = s/tℙ_2 t S_J (I + t^2
       S_J^2)^-1ℙ_1 P_s^J
       ≲s/t,

where the last line follows from the fact that S_J is self-adjoint
and therefore possesses a bounded holomorphic functional calculus with
constant independent of J. Lastly, for the third component, we have

    ℙ_3𝒫_t^J = ℙ_3 P_t^0 =
    P_t^0ℙ_3

and

    ℙ_3Π_J u = ℙ_3Π_Jℙ_1 u =
    ℙ_3Π_0ℙ_1 u

for u ∈ D(Π_J). This leads to

    ℙ_3𝒫_t^J Q_s^J   = P_t^0ℙ_3 s Π_J P_s^J
       = P_t^0ℙ_3 s Π_0ℙ_1 P_s^J
       = s/tℙ_3 t Π_0 P_t^0ℙ_1 P_s^J
       ≲s/t.

Putting everything together gives

    𝒫_t^JP_t^J - I Q_s^J≲mint/s, s/t.

This bound can then be applied to (<ref>) to give (<ref>).




 
  
 Suppose that the condition (H8Jα) is satisfied for some
 α∈ (1,2]. Then
 
    ∫^∞_0ℙ_2I - 𝒫_t^J
      P_t^J u^2dt/t≲ (α - 1)^-1b_α^J^2·u^2

for any u ∈ R Γ_J.
 

 
 It will first be proved that P_t^J u_2∈ D
   J^α - 1. Since Γ_J commutes with P_t^J we must have

    P_t^J u = Γ_J[ v; 0; 0 ]

for some v,0,0∈ D Γ_J. This then gives
P_t^J u_2 = J v. Therefore P_t^Ju_2∈
D(J^α - 1) if and only if v ∈ D(J^α). We know that
P_t^J u ∈ D(Π_J) which implies that v ∈ D(D_J^*
D_J) and therefore v ∈ D (D_J^*
  D_J)^α/2. Our hypothesis
(H8Jα) then tells us that v ∈ D(J^α)
which allows us to conclude, using the previous reasoning, that
P_t^Ju_2∈ D(J^α - 1).

   
Since (P_t^Ju)_2∈ D(J^α - 1), it follows that
   
    ℙ_2I - 𝒫_t^JP_t^Ju    = 0,t^2 J^2I + t^2 J^2^-1 P_t^J u_2,0
       = 0, g_α^tJ t^α - 1 J^α - 1
     P_t^J u_2,0,

where g_α^t : S^o_μ→ is the bounded
holomorphic function defined through

    g_α^t(z) := t^2 z^2/I + t^2 z^2 t^α -
        1√(z^2)^α - 1.

As J is self-adjoint, it follows from Corollary <ref> that J possesses a bounded holomorphic functional calculus with
constant independent of J. This, together with
(<ref>), gives

    ℙ_2I - 𝒫_t^J P_t^J u≲t^α - 1 J^α - 1P_t^J u_2.

On applying (H8Jα),

    J^α - 1P_t^J u_2   = J^α
       v
       ≤ b_α^J·D_J^*D_J^α/2 v
       = b_α^J·Π_J^α
       (v,0,0)
       ≃ b_α^J·Π_J^α - 1Π_J (v,0,0)
       = b_α^J·Π_J^α - 1
                   P_t^J u,

         where Π_J := √(Π_J^2) and in the fourth line we applied the bounded holomorphic
         functional calculus of the operator Π_J. Therefore
         
    ∫^∞_0ℙ_2I - 𝒫_t^J
       P_t^J u^2dt/t   ≲b_α^J^2·∫^∞_0t^α - 1Π_J^α - 1
       P_t^J u^2dt/t
       ≲ (α - 1)^-1b_α^J^2·u^2,

where we used the fact that Π_J is self-adjoint
and Corollary <ref> in the last line.





  
  
 Suppose that (H8Dα) is satisfied for some α∈
 (1,2]. Then
 
    ∫^∞_0ℙ_3I - 𝒫_t^J
      P_t^J u^2dt/t≲ (α - 1)^-1b_α^D^2·u^2

for all u ∈ R Γ_J.


  
First note that the left-hand side of (<ref>) can be
re-written as
 
    ∫^∞_0ℙ_3I - 𝒫_t^J
       P_t^J u^2dt/t   = ∫^∞_0ℙ_3I - P_t^0 P_t^Ju^2dt/t
       = ∫^∞_0I - P_t^0ℙ_3
       P_t^Ju^2dt/t.

It will be shown that ℙ_3 P_t^J u ∈ D
Π_0^α - 1. Since Γ_J commutes with the
operator P_t^J and u ∈ R(Γ_J), we must have
P_t^J u = Γ_J P_t^J(v,0,0) for some (v,0,0) ∈
D(Γ_J). This implies that

    ℙ_3 P_t^J u = ℙ_3Γ_J P_t^J
    (v,0,0) = ℙ_3Π_0 P_t^J(v,0,0)

and therefore ℙ_3 P_t^J u ∈ D
Π_0^α - 1 will follow from
P_t^J(v,0,0) ∈ D(Π_0^α - 1ℙ_3Π_0) which itself will follow from P_t^J(v,0,0) ∈ D (Π_0^α - 1Π_0). The bounded
holomorphic functional calculus of the operator Π_0 tells us
that D(Π_0^α - 1Π_0) = D
(Π_0^α) and it is therefore sufficient to prove that
P_t^J (v,0,0) ∈ D(Π_0^α). Since
P_t^J(v,0,0) is non-zero only in the first component, this in turn is
equivalent to proving that

    (P_t^J(v,0,0))_1∈ D(D^*D)^α/2.

This however follows directly from our hypothesis
(H8Dα) and the fact that (P_t^J(v,0,0))_1∈ D D_J^*D_J⊂ D(D_J^*D_J)^α/2. This completes the
proof of our claim that ℙ_3 P_t^Ju ∈ D
Π_0^α - 1 .

Since ℙ_3 P_t^J u ∈ D Π_0^α - 1 we must have

    I - P_t^0ℙ_3 P_t^J u = g_α^tΠ_0
    t^α - 1Π_0^α - 1ℙ_3 P_t^J u,

where g_α^t is as defined in (<ref>). 
From the bounded holomorphic functional calculus of Π_0 we then obtain

    ∫^∞_0ℙ_3I - 𝒫_t^J
       P_t^J u^2dt/t≲∫^∞_0ℙ_3 t^α - 1Π_0^α - 1
       P_t^J u^2dt/t.

On recalling that P_t^J u = Γ_J P_t^Jv, 0, 0 for some v, 0 ,
  0∈ D Γ_J,

    ∫^∞_0ℙ_3 t^α - 1Π_0^α - 1
       P_t^J u^2dt/t   = ∫^∞_0ℙ_3 t^α - 1Π_0^α - 1Γ_J P_t^J (v,0,0)^2dt/t
       = ∫^∞_0ℙ_3
                              t^α - 1Π_0^α - 1Π_0 P_t^J (v,0,0)^2dt/t.

On exploiting the bounded holomorphic functional calculus of the
operator Π_0 once more,

    ∫^∞_0ℙ_3
                              t^α - 1Π_0^α - 1Π_0 P_t^J (v,0,0)^2dt/t≲∫^∞_0t^α - 1Π_0^α P_t^J
      (v,0,0)^2dt/t.

Observe that since P_t^J(v,0,0) is non-zero only in the first entry,

    Π_0^α P_t^J (v,0,0)   = D^*D^α/2P_t^J(v,0,0)_1
       ≤
                                                                      b_α^D·D_J^*
                                                                        D_J^α/2P_t^J(v,0,0)_1
       =
                                                                      b_α^D·Π_J^α P_t^J (v,0,0).

On then applying the bounded holomorphic functional calculus of the
operator Π_J,

    b_α^D^2·∫^∞_0t^α - 1Π_J^α P_t^J (v,0,0)^2dt/t   ≲b_α^D^2∫^∞_0t^α - 1Π_J^α - 1Π_J P_t^J (v,0,0)^2dt/t
       = b_α^D^2·∫^∞_0t^α - 1Π_J^α - 1 P_t^J u^2dt/t
       ≲ (α - 1)^-1b_α^D^2·u^2,

                where we used the fact that Π_J is self-adjoint
                and Corollary <ref> in the final line.
              


Combining Propositions <ref>, <ref> and <ref>
together then gives Theorem <ref>. With this
diagonalisation in hand we can now return to our proof of Theorem
<ref>. In particular the second component of our square
function norm will now be bounded. 






Proof of Theorem <ref>.<ref>.  On splitting the second component of our square function norm from above,
   
    ∫^∞_0Θ^J,B_tℙ_2 P_t^J
       u^2dt/t≲∫^∞_0Θ^J,B_tℙ_2𝒫_t^J -
         P_t^Ju^2dt/t + ∫^∞_0Θ^J,B_tℙ_2𝒫_t^J u^2dt/t.

The uniform L^2-boundedness of the operators Θ^J,B_t
together with Propositions <ref> and
<ref> give

    ∫^∞_0Θ^J,B_tℙ_2𝒫_t^J - P_t^J u^2dt/t≲1 + b_α^J^2α - 1^-1·u^2.

It remains to bound the second term in the splitting. In order to do
so, it will first be shown that ℙ_2𝒫_t^J u ∈ D (Γ^*_J,B). From the definition
of the adjoint, ℙ_2𝒫_t^J u ∈
D(Γ^*_J,B) if and only if there exists some u' ∈ℋ such that

    ⟨Γ_J,B w, ℙ_2𝒫_t^J u ⟩ =
    ⟨ w, u' ⟩

for all w ∈ D(Γ_J,B), where Γ_J,B = B_2^*Γ_J B_1^*. For w ∈ D(Γ_J,B),

    ⟨Γ_J,B w, ℙ_2𝒫_t^J u ⟩   = ⟨ B_2^*Γ_J B_1^* w, ℙ_2𝒫_t^J u ⟩
       = ⟨Γ_J B_1^* w, B_2ℙ_2𝒫_t^J u ⟩
       = ⟨ M_J B_1^* w, B_2ℙ_2𝒫_t^J u ⟩,

where in the last line we used the fact that A_32 = 0 by
(H8Jα) and therefore B_2ℙ_2 =
ℙ_2 B_2ℙ_2.
This proves that ℙ_2𝒫_t^Ju ∈
D(Γ^*_J,B) will follow from B_2ℙ_2𝒫_t^J u ∈ D M_J^* which, in turn, will
follow from (𝒫_t^Ju)_2∈ D(J^*A_22) = D(J A_22).


Note that u ∈ R Γ_J implies
that u = Γ_J(v,0,0) for some (v,0,0)∈ D
Γ_J. Then

    𝒫_t^Ju_2   = I + t^2 J^2^-1 J v 
       = J I + t^2 J^2^-1 v.

(H8Jα) states that D(J^2) = D(J A_22J) with J^2ũ = J A_22 J ũ for ũ∈ D
J^2. Since I + t^2 J^2^-1v ∈ D J^2 we
must have I + t^2 J^2^-1v ∈ D J A_22J and
therefore 𝒫_t^J u_2∈ D J A_22. This
allows us to conclude that ℙ_2𝒫_t^Ju ∈ D
Γ_J,B^*. Moreover, from
(<ref>) we know that Γ_J,B^*ℙ_2𝒫_t^Ju = B_1 M_J^* B_2ℙ_2𝒫_t^Ju and therefore

    Γ^*_J,Bℙ_2𝒫_t^Ju   =
     B_1 M_J^* B_2ℙ_2𝒫_t^Ju
       ≲M_J^* B_2ℙ_2𝒫_t^J u
       = J A_22 J I + t^2J^2^-1v
       = J^2I + t^2 J^2^-1 v
       = J I + t^2 J^2^-1 u_2.

This together with Lemma <ref> gives

    ∫^∞_0Θ^J,B_tℙ_2𝒫_t^J u^2dt/t   = ∫^∞_0P_t^J,B t
       Γ^*_J,Bℙ_2𝒫_t^J u^2dt/t
       ≲∫^∞_0t Γ_J,B^*ℙ_2𝒫_t^J u^2dt/t
       ≲∫^∞_0t J I + t^2J^2^-1 u_2dt/t.

The theorem then follows from the fact that J is self-adjoint and
therefore satisfies square function estimates with constant
independent of J by Corollary <ref>.





 §.§ The Third Component




This section is dedicated to bounding the third component of our
square function norm and thus proving the fourth and final part of
Theorem <ref>. Specifically, it will be proved that when
(H8Dα) is satisfied for some α∈ (1,2] and either
(H8J) or (H8Jα) is satisfied the estimate

    ∫^∞_0Θ^J,B_tℙ_3 P_t^J u^2dt/t≲ c_α^J·u^2

will hold for any u ∈ R Γ_J. A similar argument to that of
<cit.> will be used, but one will need to keep
track of the effect of the projection ℙ_3.



  §.§.§ T(1)-Reduction


Our first step towards a T(1)-reduction is to use the splitting

    ∫^∞_0Θ^J,B_tℙ_3 P_t^J
      u^2dt/t≲∫^∞_0Θ^J,B_tℙ_3P_t^J -
        𝒫_t^J u^2dt/t + ∫^∞_0Θ^J,B_tℙ_3𝒫_t^J u^2dt/t.

The uniform L^2-boundedness of the operators Θ^J,B_t
and Theorem <ref> can
be applied to the first term to obtain

    ∫^∞_0Θ^J,B_tℙ_3𝒫_t^J - P_t^Ju^2dt/t≲
    c_α^J·u^2.

On recalling that ℙ_3𝒫_t^J = ℙ_3
P_t^0, this reduces the task of proving our square function
estimate to obtaining the  bound

    ∫^∞_0Θ^J,B_tℙ_3 P_t^0
      u^2dt/t≲ c_α^J·u^2.

Introduce the notation Θ̃^J,B_t to denote the operators Θ̃^J,B_t := Θ^J,B_tℙ_3.
 Let γ^J,B_t and γ̃^J,B_t denote the
 principal parts of the operators Θ^J,B_t and
 Θ̃^J,B_t respectively. That is, they are the
 multiplication operators defined through
 
    γ^J,B_t(x)w := Θ^J,B_t(w)(x)    and   γ̃^J,B_t(x)(w) := Θ^J,B_tℙ_3(w)(x),

for w ∈^N and x ∈^n. Evidently we must have
γ̃^J,B_t(x)w = γ^J,B_t(x)
ℙ_3w.

Our square function norm can be reduced to this
principal part by applying the splitting

    ∫^∞_0Θ̃^J,B_t P_t^0 u^2dt/t≲∫^∞_0Θ̃^J,B_t -
         γ̃^J,B_t A_t P_t^0 u^2dt/t + ∫^∞_0γ̃^J,B_t A_t P_t^0 u^2dt/t.

Since the operator Θ^J,B_t satisfies the conditions
of Proposition <ref>, it follows that

    ∫^∞_0Θ̃^J,B_t -
         γ̃^J,B_t A_t P_t^0 u^2dt/t   = ∫^∞_0Θ^J,B_t - γ^J,B_t
         A_tℙ_3 P_t^0u^2dt/t
       ≲∫^∞_0t ∇ℙ_3
       P_t^0u^2dt/t
       ≲∫^∞_0t Π_0 P_t^0 u^2dt/t
       = ∫^∞_0Q_t^0 u^2dt/t
       = 1/2u^2,

where the estimate ∇ℙ_3 P_t^0 u≲Π_0 P_t^0 u  follows from (H8) for the operator
Γ_0. It should be noted that in order to use (H8) we had to
use the fact that u = Γ_J v for some v ∈ D
Γ_J and therefore

    ℙ_3 P_t^0u = P_t^0ℙ_3Γ_J v =
    P_t^0ℙ_3Γ_0 v = Γ_0 P_t^0 v ∈ R Γ_0.

Our theorem has thus been reduced to a proof of the following
 square function estimate
 
    ∫^∞_0γ̃^J,B_t A_tP_t^0 u^2dt/t≲ c_α^J·u^2.

On splitting from above using the triangle inequality,

    ∫^∞_0γ̃^J,B_t A_t P_t^0
      u^2dt/t≲∫^∞_0γ̃^J,B_t A_tP_t^0 - Iu^2dt/t + ∫^∞_0γ̃^J,B_t
      A_t u^2dt/t.

Proposition <ref> states that the uniform estimate
γ̃_t^J,B A_t≲ 1  is true for all t
> 0.
Furthermore, notice that A_t^2 = A_t and ℙ_3
A_t = A_tℙ_3 for all t > 0. These facts combine
together to produce

    ∫^∞_0γ̃^J,B_t A_tP_t^0
         - Iu^2dt/t   = ∫^∞_0γ^J,B_t
       A_tℙ_3 A_tP_t^0 - I u^2dt/t
       ≲∫^∞_0ℙ_3 A_tP_t^0
         - I u^2dt/t.

According to the argument from Proposition 5.7 of
<cit.>, this final term can be bounded by

    ∫^∞_0A_tP_t^0 - Iu^2dt/t≲u^2,

since Γ_0, B_1, B_2 by hypothesis satisfies (H1)
- (H8).
For the second term in (<ref>), apply Carleson's
theorem (<cit.>) to obtain

    ∫^∞_0γ̃^J,B_t A_t u^2dt/t≲μ_𝒞·u^2,

where μ is the measure on ^n+1 defined through

    d μ(x,t) := γ̃^J,B_t(x)^2dx   dt/t

for x ∈^n and t > 0 and μ_𝒞 denotes
its Carleson norm. The proof of our theorem has thus been reduced to showing that the measure
μ is a Carleson measure with constant smaller than a multiple of c_α^J.



 


  §.§.§ Carleson Measure Estimate




 
 Our goal now is to prove the following Carleson measure
 estimate,
  
    sup_Q ∈Δ1/Q∫^l(Q)_0∫_Qγ̃^J,B_t(x)^2dx   dt/t≲ c_α^J < ∞.

  Let ℒ_3 denote the subspace

    ℒ_3 := ν∈ℒ^N∖ 0 
     : νℙ_3 = ν.


By construction, we have γ̃_t^J,B(x) ∈ℒ_3 for any t > 0 and x ∈^n since

    γ̃_t^J,B(x) ℙ_3w    = Θ^J,B_tℙ_3ℙ_3w(x) 
       = Θ^J,B_tℙ_3(w)(x) 
       = γ̃_t^J,B(x)(w).

Let σ > 0 be a constant to be determined at a later time. Let 𝒱 be a finite set consisting of ν∈ℒ_3 with
ν = 1 such that ∪_ν∈𝒱 K_ν =
ℒ_3∖ 0, where

    K_ν := ν' ∈ℒ_3∖ 0  :
    ν'/ν' - ν≤σ.

Then, in order to prove our Carleson measure estimate
(<ref>), it is sufficient to fix ν∈𝒱 and prove that

    sup_Q ∈Δ1/Q∫∫_(x,t) ∈
         R_Q
    γ̃^J,B_t(x) ∈ K_νγ̃^J,B_t(x)^2dx   dt/t≲
     c_α^J < ∞,

where R_Q := Q  [0,l(Q)). Recall the John-Nirenberg lemma for
Carleson measures as applied in <cit.> and <cit.>.


  
  Let ρ be a measure on ^n+1_+ and β > 0. Suppose
  that for every  Q ∈Δ there exists a collection Q_k_k⊂Δ of disjoint subcubes of Q such that E_Q
  := Q ∖∪_k Q_k satisfies E_Q > βQ
  and such that
  
    sup_Q ∈Δρ(E^*_Q)/Q≤ C

for some C > 0, where E^*_Q := R_Q∖∪_k R_Q_k. Then

    sup_Q ∈Δρ(R_Q)/Q≤C/β.



  
Fix Q ∈Δ and let Q_k_1_k_1 be a collection
of subcubes as in the hypotheses of the lemma. Apply the bound (<ref>) to the decomposition

    ρ(R_Q) = ρE^*_Q + ∑_k_1ρR_Q_k_1

to obtain

    ρ(R_Q) ≤ C Q + ∑_k_1ρR_Q_k_1.

For each k_1, let Q_k_1,k_2_k_2 be a collection of subcubes of
Q_k_1 that satisfy the hypotheses of the lemma. Decompose
ρ(R_Q_k_1) and once again apply (<ref>) to obtain

    ρ(R_Q)    ≤ C Q + ∑_k_1ρ(E^*_Q_k_1) +
     ∑_k_2ρ(R_Q_k_1,k_2)
       ≤ C Q + ∑_k_1 C Q_k_1 + ∑_k_1,k_2ρ(R_Q_k_1,k_2) 
       ≤ C Q + C Q1 - β + ∑_k_1,k_2ρ(R_Q_k_1,k_2).

Iterating this process and summing the resulting geometric series
gives (<ref>).
 

 With this tool at our disposal, the proof of our theorem is reduced to the
following proposition.

 
  
 There exists β > 0 and σ > 0 that will satisfy the
 following conditions. For every ν∈𝒱 and Q ∈Δ, there is a collection Q_k_k⊂Δ of
 disjoint subcubes of Q such that E_Q,ν = Q ∖∪_k Q_k
 satisfies E_Q,ν > βQ and such that
 
    sup_Q ∈Δ1/Q∫∫_(x,t) ∈
         E^*_Q,ν
    γ̃^J,B_t(x) ∈ K_νγ̃^J,B_t(x)^2dx  
     dt/t≲ c_α^J < ∞,

where E^*_Q,ν := R_Q∖∪_k R_Q_k. Moreover, β
and σ are entirely independent of the conditions (H8Dα),
(H8J) and (H8Jα).
 

 For now, fix ν∈𝒱 and Q ∈Δ. Let w^ν,
 ŵ^ν∈^N with ŵ^ν =
 w^ν = 1 and ν^*ŵ^ν = w^ν. To
 simplify notation, when superfluous, this dependence will be kept
 implicit by defining w := w^ν and ŵ :=
 ŵ^ν. Notice that since ν satisfies ν = νℙ_3, w must satisfy ℙ_3 w = w.



  For ϵ > 0  the function f^w_Q,
  ϵ can be defined in an identical manner to
<cit.>. Specifically, let η_Q : ^N→ [0,1] be a smooth cutoff function equal to 1 on 2 Q,
with support in 4 Q and with ∇η_Q_∞≤1/l, where l := l(Q). Then define w_Q := η_Q· w and

    f^w_Q, ϵ := w_Q - ϵ  l i Γ_JI + ϵ l
    i Π_J,B^-1 w_Q = I + ϵ l i Γ^*_J,BI +
    ϵ l i Π_J,B^-1 w_Q.


 
  
 There exists a constant C > 0, independent of (H8Dα), (H8J)
 and (H8Jα), that
 satisfies f^w_Q, ϵ≤ C Q^1/2
 and
 
    _Qℙ_3 f^w_Q, ϵ - w≤ C ·ϵ^1/2,

for any ϵ > 0.



  The first claim follows from
  
    f^w_Q,ϵ   ≲w_Q + ϵ l i
       Γ_JI + ϵ l i Π_J,B^-1 w_Q
       ≲Q^1/2 + ϵ l i Π_J,BI
         + ϵ l i Π_J,B^-1w_Q
       ≲Q^1/2.

  On recalling that w is zero in the first two components,
  
    _Qℙ_3 f^w_Q,ϵ - w^2   =
     _Qℙ_3ϵ l i Γ_JI + ϵ
         l i Π_J,B^-1 w_Q^2
       = _Qϵ l i Γ_0I + ϵ l i Π_J,B^-1w_Q^2.

At this point, apply Lemma 5.6 of <cit.> to the
operator Υ = Γ_0 to obtain

    _Qϵ l i Γ_0I + ϵ l i
         Π_J,B^-1w_Q^2   ≲ϵ l^2/l_QI + ϵ l i Π_J,B^-1
         w_Q^2^1/2
               ·_QΓ_0I + ϵ l i Π_J,B^-1 w_Q^2^1/2
       ≲ϵ_Qϵ l i Γ_0I +
           ϵ l i Π_J,B^-1 w_Q^2^1/2
       ≤ϵ_Qϵ l i Γ_JI +
           ϵ l i Π_J,B^-1 w_Q^2^1/2
       ≲ϵ,

where the inequality Γ_0 v≤Γ_J v
for v ∈ D Γ_J follows trivially from the matrix form
of Γ_0 and Γ_J.
 

  
  
There exists a constant D > 0, independent of (H8Dα), (H8J)
and (H8Jα), such
that

    ∫∫_R_QΘ^J,B_t f^w_Q,ϵ(x)^2dx   dt/t≤ D Q/ϵ^2.



  
  First observe that
  
    Θ^J,B_t f^w_Q,ϵ   =  P_t^J,B t Γ^*_J,BI + ϵ l i Γ^*_J,BI + ϵ l i
      Π_J,B^-1 w_Q
       = t/ϵ l P_t^J,Bϵ l Γ^*_J,BI
      + ϵ l i Π_J,B^-1 w_Q.

Therefore

    ∫^l_0∫_QΘ^J,B_t
       f^w_Q,ϵ(x)^2dx   dt/t   = ∫^l_0t/ϵ l^2∫_QP_t^J,Bϵ l
       Γ^*_J,BI + ϵ l i Π_J,B^-1 w_Q^2
     dx  dt/t
       ≲∫^l_0t/ϵ l^2ϵ l
     i Γ^*_J,BI + ϵ l i Π_J,B^-1 w_Q^2dt/t
       ≲Q/ϵ l^2∫^l_0 t   dt 
       ≃Q/ϵ^2.

 


 From this point forward, with C as in Lemma <ref>, set
 ϵ := 1/4 C^2 and introduce the notation f^w_Q := f^w_Q,
   ϵ. With this choice of ϵ it must be true that
 
    _Qℙ_3f^w_Q - w≤1/2.

That is,

    1 - 2 Re⟨_Qℙ_3 f^w_Q, w ⟩   =
     w^2 - 2 Re⟨_Qℙ_3 f^w_Q, w ⟩
       ≤_Qℙ_3 f^w_Q - w^2
       ≤1/4.

On rearranging we find that

    Re⟨_Qℙ_3f^w_Q, w ⟩≥1/4.



 
 In this context, Lemma 5.11 of <cit.> will take on the below form.

  
   
   There exists β, c_1, c_2 > 0 and a collection Q_k of dyadic cubes of Q
   such that E_Q,ν > βQ and such that
   
    Re⟨ w, _Q'ℙ_3 f^w_Q⟩≥ c_1   and   _Q'ℙ_3f^w_Q≤ c_2

for all dyadic subcubes Q' ∈Δ of Q which satisfy R_Q'∩ E^*_Q, ν≠∅. Moreover, β, c_1 and
c_2 are independent of (H8Dα), (H8J), (H8Jα), Q, σ and ν.


The proof of this statement follows in an identical manner to the
argument in <cit.>. If we set σ = c_1/2 c_2, then the following pointwise estimate can be
deduced.

 
  
 If (x,t) ∈ E^*_Q, ν and γ̃_t^J,B(x) ∈
 K_ν then
 
    γ̃_t^J,B(x) A_t f^w_Q(x)≥1/2
       c_1γ̃^J,B_t(x).

 

 
   First observe that

    νA_t f^w_Q(x)   ≥Re⟨ŵ,
     νA_t f^w_Q(x)⟩
       = Re⟨ w , A_t f^w_Q(x) ⟩
       = Re⟨ w, A_tℙ_3 f^w_Q(x) ⟩
       ≥ c_1.

Then

    γ̃^J,B_t(x)/γ̃^J,B_t(x)A_t f^w_Q(x)   =  γ̃^J,B_t(x)/γ̃^J,B_t(x)A_tℙ_3 f^w_Q(x)
       ≥νA_t f^w_Q(x) -
     γ̃^J,B_t(x)/γ̃^J,B_t(x)
       - νA_tℙ_3 f^w_Q(x)
       ≥ c_1 - σ c_2
       = 1/2 c_1.



Proof of Proposition <ref>. From the pointwise bound of the previous lemma,

    ∫∫_(x,t) ∈ E^*_Q,ν
    γ̃_t^J,B(x) ∈ K_νγ̃_t^J,B(x)^2dx   dt/t≲∫∫_R_Qγ̃^J,B_t(x) A_t f^w_Q(x)^2dx    dt/t.

At this stage we can begin to unravel our square function norm,

    ∫∫_R_Qγ̃^J,B_t(x) A_t
       f^w_Q(x)^2dx   dt/t   ≲∫∫_R_QΘ^J,B_t
       f^w_Q(x) - γ̃^J,B_t(x)A_t f^w_Q(x)^2dx   dt/t
                + ∫∫_R_QΘ^J,B_t f^w_Q(x)^2dx   dt/t.

 Lemma <ref> states that the final term in the above
estimate will be bounded from above by a multiple of Q. This reduces the task of
proving the proposition to bounding the first term of the above
splitting.
Recall that f^w_Q can be expressed in the form

    f^w_Q := w_Q - u^w_Q,

where u^w_Q∈ R Γ_J is given by

    u^w_Q := ϵ l i Γ_JI + ϵ l i
      Π_J,B^-1 w_Q.

An application of the triangle inequality then leads to

    ∫∫_R_Q   Θ^J,B_t f^w_Q(x) -
      γ̃^J,B_t(x) A_t f^w_Q(x)^2dx  
      dt/t
               ≲∫∫_R_QΘ^J,B_t w_Q(x) -
      γ̃^J,B_t(x) A_t w_Q(x)^2dx  
      dt/t
                    + ∫∫_R_QΘ^J,B_t u^w_Q(x) -
      γ̃^J,B_t(x) A_t u^w_Q(x)^2dx   dt/t.

On noticing that for every x ∈ Q and 0 < t < l(Q)

    Θ^J,B_t w_Q(x) - γ̃^J,B_t(x) A_t
     w_Q(x)    = Θ^J,B_t w_Q(x) - Θ^J,B_tA_t
       w_Q(x)(x) 
       = Θ^J,B_tη_Q - 1 w(x),

it is clear that the first term in (<ref>) can be handled
in  an identical manner as in the proof
of Proposition 5.9 from <cit.>. Specifically, 
since  suppη_Q - 1 w∩ 2 Q = ∅,
the off-diagonal estimates of the operator Θ^J,B_t lead to

    ∫_QΘ^J,B_tη_Q - 1 w(x)^2 dx
    ≲t Q/l,

which implies that

    ∫∫_R_QΘ^J,B_t w_Q(x) -
      γ̃^J,B_t(x) A_t w_Q(x)^2dx   dt/t≲Q.

As for the second term in (<ref>),

    ∫∫_R_Q   Θ^J,B_t u^w_Q -
       γ̃^J,B_t(x) A_t u^w_Q(x)^2dx  
     dt/t
               ≲∫∫_R_QΘ^J,B_tI -
       P_t^Ju^w_Q(x)^2dx   dt/t
                    + ∫∫_R_QΘ^J,B_t P_t^J u^w_Q(x) -
      γ̃^J,B_t(x) A_t u^w_Q(x)^2dx
        dt/t.

Since u_Q^w∈ R Γ_J, Corollary <ref> gives

    ∫∫_R_QΘ^J,B_tI - P_t^J
       u^w_Q^2dx   dt/t   ≲u^w_Q^2
       ≲Q.

 For the remaining term in (<ref>),

    ∫∫_R_Q   Θ^J,B_t P_t^J u^w_Q(x) -
       γ̃^J,B_t(x) A_t u^w_Q(x)^2dx  
     dt/t
               ≲∫∫_R_QΘ^J,B_tI -
       ℙ_3 P_t^J u^w_Q^2dx   dt/t
                    + ∫∫_R_QΘ̃^J,B_t P_t^J
      u^w_Q(x) - γ̃^J,B_t(x) A_t u^w_Q(x)^2dx   dt/t.

Since we have already proved the boundedness of the first and second components,

    ∫∫_R_QΘ^J,B_tI - ℙ_3
       P_t^J u^w_Q^2dx   dt/t   ≲ c_α^J·u^w_Q^2
       ≲ c_α^J·Q.

For the second term in (<ref>),

    ∫∫_R_Q   Θ̃^J,B_t  P_t^J
       u^w_Q(x) - γ̃^J,B_t(x) A_t u^w_Q(x)^2dx   dt/t
               ≲∫∫_R_QΘ̃^J,B_t P_t^J u^w_Q(x) -
     γ̃^J,B_t(x) A_t P_t^J u^w_Q(x)^2dx   dt/t
                    + ∫∫_R_Qγ^J,B_t(x) ℙ_3 A_t
      P_t^J -  A_t u^w_Q(x)^2dx   dt/t.

To bound the first term on the right-hand side of the above estimate
notice that

    Θ̃^J,B_t P_t^J u^w_Q(x) -
     γ̃^J,B_t(x) A_t P_t^J u^w_Q(x) =
     Θ^J,B_t - γ^J,B_t A_tℙ_3
     P_t^J u^w_Q(x).

 Theorem <ref> then allows us to diagonalise our
 P_t^J operators in the first term of
 (<ref>) to get
 
    ∫^l(Q)_0Θ^J,B_t - γ^J,B_t
        A_tℙ_3 P_t^J u^w_Q^2_L^2Qdt/t≲
    c_α^JQ + ∫^∞_0Θ^J,B_t
        - γ^J,B_t A_tℙ_3𝒫_t^J
      u^w_Q^2dt/t.

From Proposition
<ref> we know that

    ∫^∞_0Θ^J,B_t - γ_t^J,B
         A_tℙ_3𝒫_t^J u^w_Q^2dt/t   ≲∫^∞_0t ∇ℙ_3𝒫_t^J u^w_Q^2dt/t
       = ∫^∞_0t ∇ℙ_3 P_t^0
       u^w_Q^2dt/t
       ≲∫^∞_0t Π_0 P_t^0 u^w_Q^2dt/t
       = ∫^∞_0Q_t^0u^w_Q^2dt/t
       ≲Q,

where in the third line we applied (H8) for the operators Γ_0,B_1, B_2. It should be noted that in order to use
(H8), we had to use the fact that

    ℙ_3 P_t^0 u^w_Q   = P_t^0ℙ_3
     u^w_Q = P_t^0ℙ_3ϵ l i Γ_J (I +
     ϵ l i Π_J,B)^-1 w_Q
       = P_t^0ℙ_3ϵ l i Γ_0 (I + ϵ l i
     Π_J,B)^-1 w_Q = ϵ l i Γ_0 P_t^0 (I +
     ϵ l i Π_J,B)^-1 w_Q∈ R (Γ_0).

It remains to bound the second term in (<ref>),

    ∫∫_R_Qγ^J,B_t(x) ℙ_3 A_tP_t^J - I u^w_Q(x)^2dx   dt/t = ∫∫_R_Qγ^J,B_t A_tℙ_3 A_tP_t^J - I u^w_Q(x)^2dx   dt/t

On noting the uniform L^2-boundedness of the γ^J,B_t
A_t operators and applying the triangle inequality,

    ∫∫_R_Qγ^J,B_t A_tℙ_3 A_tP_t^J - I u^w_Q(x)^2dx   dt/t   ≲∫_0^∞∫_^nℙ_3 A_tP_t^J - I
      u^w_Q(x)^2dx   dt/t
       ≲∫_0^∞∫_^nℙ_3 A_tP_t^J -
        𝒫_t^J u^w_Q(x)^2dx   dt/t
            + ∫_0^∞∫_^nℙ_3 A_t𝒫_t^J - I
      u^w_Q(x)^2dx   dt/t.

Applying Theorem <ref> and recalling that ℙ_3𝒫_t^J = ℙ_3 P_t^0,

    ∫∫_R_Qγ^J,B_t A_tℙ_3 A_tP_t^J - I u^w_Q(x)^2dx   dt/t   ≲
    c_α^J·u^2 +  ∫_0^∞∫_^nℙ_3 A_tP_t^0 - I
      u^w_Q(x)^2dx   dt/t.

From the proof of Proposition 5.7 of <cit.> we
know that

    ∫_0^∞∫_^nA_tP_t^0 - Iu^w_Q(x)^2dx   dt/t≲Q,

allowing us to finally conclude our proof.




   






§ APPLICATIONS




Our non-homogeneous framework will now be applied to three
different contexts. We begin with the case that serves as the primary
motivation for this article,
the scalar Kato square root problem with zero-order
potential.



 §.§ Scalar Kato with Zero-Order Potential

 



 
Theorem <ref>,
the promised result of the introductory section, will now be
proved. Fix V ∈𝒲_α for some α∈ (1,2].
 Brand the definition of the operators Γ_J, B_1 and B_2 to be as
follows. Define our Hilbert space to be

    ℋ := L^2^n⊕ L^2^n⊕ L^2(^n;^n),

for some n ∈^*. Set J = V^1/2 and D = ∇. 
 Our operator Γ_J is
then given by

    Γ_J = Γ_V^1/2 = [       0       0; ∇_V^1/2       0 ] =  [     0     0     0; V^1/2     0     0;     ∇     0     0 ],

 defined on the dense domain H^1,V^n⊕ L^2^n⊕
 L^2^n;^n, where H^1,V^n is as defined
 in the introductory section.
The density of H^1,V^n in L^2^n follows
from V ∈ L^1_loc(^n).

Let A ∈ L^∞^n; ℒ^n be a
matrix-valued multiplication operator and suppose that the Gårding
inequalities (<ref>) and (<ref>) are 
satisfied with constants κ^A > 0 and κ_V^A > 0 respectively.
Define our perturbations B_1 and B_2 through
 
    B_1 = I      and
         B_2 := [ I 0; 0 Â ] :=  [          I          0          0;          0 e^i ·arg V          0;          0          0          A ].

Our perturbed Dirac-type operator then becomes
 
    Π_B,V^1/2 := Γ_V^1/2 +
    Γ^*_V^1/2[ I 0; 0 Â ] = [          0 ∇_V^1/2^*Â;    ∇_V^1/2          0 ].

The square of our perturbed Dirac-type operator is then given by

    Π_B,V^1/2^2 = [ ∇^*_V^1/2Â∇_V^1/2                 0;                 0 ∇_V^1/2∇^*_V^1/2Â ] = [   - div (A ∇) + V                 0;                 0 ∇_V^1/2∇^*_V^1/2Â ].

It is clear from the form of our operator Γ_0 and the fact
that A satisfies (<ref>) that the operators Γ_0, B_1, B_2 satisfy (H1) - (H8). Similarly, since
A and V satisfy (<ref>), it follows that Γ_J, B_1, B_2 satisfy the properties (H1) - (H6).




  
  For V ∈𝒲_α, the conditions (H8Dα) and
(H8Jα) are both satisfied and
 c_α^J≲1 + V_α^2α - 1^-1.


  
  The condition V ∈𝒲_α tells us that
  
    V^α/2u_2 +
        (-Δ)^α/2u_2≤V_αV - Δ^α/2u_2

  for all u ∈ D(V - Δ). Let's first prove that
  (H8Dα) is satisfied. In order to prove this
  condition, it is sufficient to show that D (V -
    Δ)^α/2⊂ D
  (-Δ)^α/2 and
  
    (-Δ)^α/2u_2≤V_α(V - Δ)^α/2u_2

  for all u ∈ D (V - Δ)^α/2. Fix u
  ∈ D (V - Δ)^α/2. Since
  D(V - Δ) is a core for (V -
  Δ)^α/2, there must exist some u_n_n
    = 1^∞⊂ D(V - Δ) with
  
    u_n - u_2 + (V - Δ)^α/2(u_n - u)_2 0.

  We then have for n,   m ∈,
  
    u_n - u_m_2 + (-Δ)^α/2(u_n - u_m)_2≤u_n - u_m_2 + V_α(V -
      Δ)^α/2 (u_n - u_m)_2.

This proves that u_n_n = 1^∞ is Cauchy in the
graph norm of (-Δ)^α/2. The sequence u_n_n = 1^∞ must therefore 
converge to some ũ∈ D (-Δ)^α/2
in the graph norm of (-Δ)^α/2,

    u_n - ũ_2 + (-Δ)^α/2 (u_n -
      ũ)_2 0.

This combined with (<ref>) shows that u =
ũ and therefore u ∈
D(-Δ)^α/2. Moreover, we have that

    (-Δ)^α/2u_2   = lim_n →∞(-Δ)^α/2u_n_2
       ≤V_αlim_n →∞(V -
       Δ)^α/2u_n_2
       = V_α(V - Δ)^α/2u_2,

completing the proof of (H8Dα) with
b^D_α≲V_α. An identical proof can be
used to obtain the condition (H8Jα) with
constant b^J_α≲V_α.
 

Combining the above Lemma with Corollary <ref> completes the proof of Theorem <ref>.









 §.§ The Class 𝒲



Define the potential class

    𝒲 := ⋃_α∈ (1,2]𝒲_α.

It has so far been proved that the Kato estimate holds for any potential
in the class 𝒲.
At this stage, however,
the unperturbed condition V ∈𝒲 remains in quite an
abstract form. It will therefore be instructive to unpack this
condition and compare 𝒲 with other commonly used classes
of potentials. It is first interesting to note that 𝒲_1
is the collection of all potentials with no additional restrictions.

 
  
 For any locally integrable  V: ^n→ we have V_1≤
 2. That is, 𝒲_1 = L^1_loc(^n).


  
  We have
  
    (-Δ)^1/2u_2^2   = ⟨
     (-Δ)^1/2u, (-Δ)^1/2 u ⟩_2 = ⟨
     (-Δ)u, u ⟩_2
       ≤⟨ - Δ u, u ⟩_2 + ⟨V u, u ⟩_2
       = ⟨ (V - Δ)^1/2u, (V -
     Δ)^1/2u ⟩_2 = (V - Δ)^1/2u_2^2

for all u ∈ D(V - Δ). The second estimate follows
in a similar manner from the non-negativity of (-Δ).
 

Using this as an endpoint, it can then be proved using an interpolation style argument that the
potential classes 𝒲_α_α∈ [1,2]
form a decreasing collection.
 
 
  
 Suppose that the potential V : ^n→ is in
 𝒲_α for some α∈ (1,2]. Then V ∈𝒲_β for any β∈ [1,α] with
 
    V_β≤  2 V_α^β - 1/α - 1.

Therefore for β≤α

    𝒲_β⊃𝒲_α.




 Assume that V ∈𝒲_α for some α∈ (1,2]. It
will be shown through the Hadamard three-lines theorem that V ∈𝒲_β for
β∈ [1,α] with the constant given in
(<ref>).

Let St :=  z ∈ : 0 < Re  z < 1 and set
θ := β - 1/α - 1. Fix u ∈ D(V - Δ) and define f : St→ to be the function given by

    f(z) := V^1/2 + α - 1/2 zV - Δ^α - 1/2θ - zu_2.

f is holomorphic on St and continuous on the closed strip
St. In order to apply the three-lines theorem, it must
first be proved that f is bounded on
St. For z = s + it where 0 ≤ s ≤ 1 and t ∈ we have

    f(z)     = V^1/2 + α - 1/2s+ i tV - Δ^β - 1/2 - α -
             1/2(s + i t)u_2
       = V^1/2 + α - 1/2sV - Δ^-1/2 - α - 1/2sV - Δ^-α - 1/2 i tV -
       Δ^β/2u_2.

The function

    s ↦V^1/2 + α - 1/2sV - Δ^-1/2 - α - 1/2sv_2

is continuous on [0,1] for v ∈ L^2^n. This, together with (<ref>)
then gives

    f(z)    ≲V - Δ^-α -
             1/2i tV - Δ^β/2u_2
       ≤V - Δ^β/2u_2,

where we used the fact that V - Δ^i a is a
contraction operator on L^2^n for a ∈. This
demonstrates that f is bounded on St.


For t ∈, Proposition <ref> implies that

    f(it)    = V^1/2 + α - 1/2 i tV - Δ^α - 1/2θ - i
           tu_2
       = V^1/2V -
         Δ^α - 1/2θ - i t u_2
       ≤V - Δ^1/2V -
         Δ^α - 1/2θ - i tu_2
       ≤V - Δ^β/2u_2.

We also have

    f(1 + i t)    = V^α/2 + α -
             1/2 i tV - Δ^α - 1/2θ - 1 - i tu_2
       = V^α/2V -
         Δ^α - 1/2θ - 1 - i tu_2
       ≤V_αV - Δ^α/2V - Δ^α - 1/2θ - 1 - i
         tu_2
       ≤V_αV - Δ^α/2 +
        α - 1/2θ - 1u_2
       = V_αV - Δ^β/2u_2.

The Hadamard three-lines theorem then gives the bound

    f(θ) = V^β/2u_2≤V_α^β - 1/α - 1V
      - Δ^β/2u_2.

A similar argument can be applied to obtain the bound

    - Δ^β/2u_2≤V_α^β - 1/α - 1V
      - Δ^β/2u_2

for all u ∈ D(V - Δ), one must simply remember that
the imaginary powers of the positive self-adjoint operator
-Δ are contraction operators on
L^2^n. Combining (<ref>) and
(<ref>) then gives V_β≤ 2
V_α^β - 1/α - 1.


Recall that a non-negative potential V ∈ L^q_loc(^n) for index
1 < q < ∞ is said to
belong to the reverse Hölder class RH_q if there exists C > 0 such that

    1/B∫_B V^q  dx^1/q≤ C
    1/B∫_B V   dx

for every ball B ⊂^n. For V ∈ RH_q, the condition V_α < ∞ for
α = 2 was proved in <cit.> for q ≥n/2 and n ≥ 3. This
was later improved to q
≥ 2 and arbitrary dimension in <cit.>. Note
that in <cit.> the estimate Δ u_2 +
V u_2≲(V -Δ)u_2 was
proved for u ∈ C^∞_0(^n). However, this estimate can
then be extended to all of D(V - Δ) since V∈
L^2_loc(^n) implies that C^∞_0(^n) is a core
for (V - Δ) (c.f. <cit.>).





  
  Let V ∈ L^1_loc(^n) and suppose that V∈
  RH_q for some q ≥ 2. Then V ∈𝒲_2⊂𝒲.




Notice that since the absolute value of any polynomial is contained in
RH_q for any q ∈ (1,∞) we automatically obtain the following corollary.
 



  
  For any polynomial P, we have P ∈𝒲_2. As a result, the Kato estimate holds for any polynomial with range
  contained in S_μ+ for some μ∈ [0,π/2).
  


The ensuing proposition demonstrates that
the inclusion of the reverse Hölder potentials in
𝒲_2 is strict, at least in dimension n > 4.

 
  
 For n > 4,
 
    L^n/2^n⊂𝒲_2.

 

   
   Fix V ∈ L^n/2^n. For ε > 0, the
   resolvent (V + ε - Δ)^-1 is well-defined as a bounded
   operator on L^2(^n). Hölder's inequality followed by
    the fact that (V + ε - Δ)^-1u ≤
    (-Δ + ε)^-1u for any u ∈ L^2 and then
    finally the uniform boundedness of (-Δ + ε)^-1
    from L^2 n/n - 4 to L^2 (c.f. <cit.>) produces
   
    (V+ ε) V + ε -
         Δ^-1 u_L^2(B(0,N))
                   ≤V + ε_L^n/2(B(0,N))·V + ε -
         Δ^-1u_L^2n/n - 4(B(0,N))
                   ≤V + ε_L^n/2(B(0,N))·-Δ + ε^-1u_L^2n/n - 4(B(0,N))
                   ≲V + ε_L^n/2(B(0,N))u_2

for any N > 0, where B(0,N) ⊂^n is the open ball of radius N centered
at the origin. Therefore,

    (V + ε)u_L^2(B(0,N))   ≲V +
      ε_L^n/2(B(0,N))·(V + ε -
      Δ)u_2
       ≤V + ε_L^n/2(B(0,N))·(V -Δ)u_2 + εu_2

for all u ∈ D(V + ε - Δ) = D(V -
Δ). Letting ε→ 0 followed by N
→∞ gives the estimate

    V u_2≲V_n/2·(V - Δ )u_2

for all u ∈ D(V - Δ). The triangle inequality then
yields the bound (-Δ)u_2≲(V -
  Δ)u_2 and thus V ∈𝒲_2.
 

 The above statements demonstrate clearly that the class of potentials
 𝒲_2 is quite large. In light of Proposition
 <ref>, however, it is also evident that 𝒲_2 is the smallest
 class out of the collection 𝒲_α_α∈ (1,2]. One can then neatly surmise that
 𝒲, the class of all potentials for which the Kato
 estimate holds, is certain to be quite large itself. 





 

 §.§ Systems with Zero-Order Potential

 

 Fix m ∈^* and A ∈ L^∞^n; ℒ^n⊗^m.
 Let V : ^n→ℒ^m be a measurable matrix-valued function
 contained in L^1_loc(^n;ℒ(^m)) with V(x)
 normal for almost every x ∈^n.  V can
 be viewed as a densely defined closed multiplication operator on
 L^2^n;^m with domain
 
    D V =  u ∈ L^2^n;^m : V · u ∈ L^2^n;^m.

Similar to the scalar case, one can define forms 𝔩^A
and 𝔩_V^A respectively through

    𝔩^Au,v := ∫_^n⟨ A(x) ∇ u(x),
    ∇ v(x) ⟩_^n⊗^m  dx

for u,   v ∈ H^1^n;^m and

    𝔩_V^Au',v' := 𝔩^Au',v' +
    ∫_^n⟨ V(x) u'(x), v'(x) ⟩_^m  dx

for u' and v' contained in

    H^1,V^n;^m := H^1^n;^m∩ D(V^1/2),

where V(x) := √(V(x)^*V(x)) for x ∈^n. The density of H^1,V(^n;^m) in
L^2(^n;^m) follows from the fact that
C^∞_0(^n;^m) ⊂ H^1,V(^n;^m).
Assume that the forms 𝔩^A and
𝔩_V^A satisfy the system equivalents of the Gårding inequalities (<ref>) and
(<ref>) with constants κ^A > 0 and
κ_V^A > 0
respectively. Then 𝔩^A and 𝔩_V^A will
both have a unique associated maximal accretive operator, L
and L + V.


In the article <cit.>, the Kato square
root property √(L) u_2≃∇ u_2 was proved
for elliptic systems without potential. Using the non-homogeneous
machinery that we have developed, the corresponding property can be
proved for the operator with potential. Define V_α and
𝒲_α(^n;ℒ(^m)) for α∈ [1,2] to be the system
analogues of the corresponding scalar objects. In the below theorem, our non-homogeneous
framework will be applied to determine the domain of √(L + V) for normal potentials.




 
  
  Suppose that V ∈𝒲_α(^n;
  ℒ(^m)) for some α∈ (1,2] and that V(x) is a normal matrix for almost every x ∈^n.  Suppose that the system equivalents of the Gårding inequalities (<ref>) and
(<ref>) are both satisfied with constants κ^A
and κ_V^A respectively.
  Then there must exist some C_V >
0 such that

    C_V^-1V^1/2 u_2 +
      ∇ u_2≤√(L + V) u_2≤ C_VV^1/2 u_2 + ∇ u_2

for all u ∈ H^1,V^n;^m. Moreover, the constant
C_V depends on V and α through

    C_V = C̃_V (α - 1)^-1 (1 + V_α^2),

where C̃_V only depends on V through κ_A^V and
is independent of α.
 

 
   The polar decomposition theorem asserts the existence of some U :
   ^n→ℒ(^m), with U(x) unitary for all
   x ∈^n, such that
   
    V(x) = U(x) V(x)

for all x ∈^n. As
V(x) is normal, the matrices U(x) and V(x) are well-known
to commute. We therefore have the decomposition

    V(x) = V(x)^1/2 U(x) V(x)^1/2

for almost every x ∈^n.
Set

    D := ∇ : H^1^n;^m⊂ L^2^n ; ^m→
    L^2^n;^n⊗^m

and

    J := V^1/2 : D(V^1/2) ⊂
    L^2^n;^m→ L^2^n;^m,

both defined as operators on L^2^n;^m. Define the
perturbation matrices

    B_1 := I      and      B_2 = [ I 0 0; 0 U 0; 0 0 A ].

It is not too difficult to see that the operators Γ_0,
B_1, B_2 will satisfy conditions (H1) - (H8) and Γ_J, B_1, B_2 will satisfy (H1) - (H6). Indeed, the only
non-trivial condition for both sets of operators is (H2) and this
follows from the respective Gårding inequalities
(<ref>) and (<ref>). It is also clear from
the fact that V ∈𝒲_α(^n; ℒ(^m)) that (H8Dα) and
(H8Jα) will both be satisfied. This follows from an argument
identical to that of Lemma <ref>.  The Kato estimate then follows
from Corollary <ref> with constant C̃_V (α
- 1)^-1(1 + V_α^2). It should be noted that
(<ref>) was needed so that we would have L^J_B =
L + V.
 


 
 In analogy to the scalar case, it is quite likely that a similar reverse
Hölder type condition will be sufficient to imply the boundedness
of the operator VV - Δ^-1 for m > 1. However,
as far as the author is aware, this is still an open problem for m > 1. What is
apparent is that the condition that the potential belongs to
L^n/2 will once again be sufficient to imply that it
belongs to 𝒲_2. The following proposition has an
identical proof to that of Proposition <ref>.



  
  For n > 4 and m ∈^*,
  
    L^n/2^n ; ℒ^m⊂𝒲_2(^n; ℒ(^m)).






 

 §.§ First Order Potentials

 


 Let b : ^n→^n be contained in
 L^1_loc^n;^n and A ∈ L^∞^n;ℒ(^n). Suppose that A satisfies the standard Gårding inequality

    Re∫_^n⟨ A(x) ∇ u(x), ∇ u(x)
    ⟩_^n  dx ≥κ^A·∇ u_2^2

for all u ∈ H^1^n;, for some κ^A > 0. Consider the accretive sesquilinear form
 
    𝔥^A_b[u,v] := ⟨ A ∇ u, ∇ v ⟩_2
    + ⟨ (∇ + b) u, (∇ + b)v ⟩_2

defined on the dense subspace H^1_b(^n) :=  u ∈
H^1(^n) : b u∈ L^2(^n) ⊂ L^2(^n). 


  The form 𝔥^A_b is both continuous and closed.


  
  Let's first prove that 𝔥^A_b is continuous. It must
  be shown that 𝔥^A_b[u,v]≲u_𝔥^A_bv_𝔥^A_b for
  all u,   v ∈ H^1_b,
  where u_𝔥^A_b := √(Re𝔥^A_b[u,u] + u_2^2). On consecutively
  applying the boundedness of A and then the accretivity of A,
  
    𝔥^A_b[u,v]   = ⟨ A ∇ u, ∇ v
       ⟩_2 + ⟨ (∇ + b)u, (∇ + b) v ⟩_2
       ≲∇ u_2∇ v_2 + (∇ +
       b)u_2(∇ + b) v_2
       ≲√(Re ⟨ A ∇ u, ∇ u
       ⟩_2)√(Re ⟨ A ∇ v, ∇ v
       ⟩_2) + (∇ + b)u_2(∇ + b)v_2
       ≲u_𝔥^A_bv_𝔥^A_b.

Let's now prove that 𝔥^A_b is closed. That is, it
must be proved that H^1_b(^n) is complete under the norm
·_𝔥^A_b. From the boundedness of A,

    u_𝔥^A_b   = √(Re𝔥^A_b[u,u] + u_2^2)
       = √(Re⟨ A ∇ u, ∇ u ⟩_2 +
       (∇ + b)u_2^2 + u_2^2)
       ≲√(∇ u^2_2 + b u^2_2 +
       u_2^2)
       ≲u_H^1_b,

where u_H^1_b := u_2 + ∇ u_2 +
b u_2. Conversely, the Gårding inequality for A implies
that

    u_𝔥^A_b   = √(Re⟨ A
         ∇ u, ∇ u ⟩_2 + (∇ + b)u_2^2 +
       u_2^2)
       ≳∇ u_2 + (∇ + b) u_2 + u_2
       ≥∇ u_2 + 1/2(∇ + b)u_2 +
     u_2
       ≥∇ u_2 + 1/2b u_2 - 1/2∇ u_2 + u_2
       ≃u_H^1_b.

This shows that the norm ·_𝔥^A_b is
equivalent to ·_H^1_b on H^1_b. Since
H^1_b is known to be complete under the norm ·_H^1_b it then follows that it must
also be complete under ·_𝔥^A_b.
 

The previous lemma implies, in particular, that there exists a maximal
accretive operator ℒ^A_b associated with this form
(c.f. <cit.>). This operator will be denoted by

    ℒ^A_b = (∇ + b)^* (∇ + b) - div(A ∇).

 Define the Hilbert space to be
 
    ℋ := L^2^n⊕ L^2^n;^n⊕ L^2^n;^n.

Then set

    J := ∇ + b : L^2^n; → L^2^n;
      ^n   and    D := ∇ : L^2^n ; → L^2^n;^n.

Also, let B_1 = I as usual and

    B_2 = [ I 0 0; 0 I 0; 0 0 A ].

Then the operator L_B^J as in Corollary <ref> is  given
by L_B^J = ℒ^A_b. Since A satisfies the standard
Gårding inequality it follows that Γ_0, B_1,
B_2 and Γ_J, B_1, B_2 will both satisfy
(H2). This in turn implies that Γ_0, B_1, B_2
satisfies (H1) - (H8) and Γ_J, B_1, B_2 satisfies
(H1) - (H6). The non-homogeneous framework, in the form of Corollary <ref>, applied to these operators
then produces the following theorem.

 
  
 Suppose that D ∇ + b^*∇ + b -
  Δ⊂ D (-Δ) ∩ D(∇ + b^*∇ + b) and there exists some c_b > 0 such that
 
    Δ u_2≤ c_b∇ + b^*∇ +
        b - Δu_2

for all u ∈ D ∇ + b^*∇ + b -
  Δ. Then there exists some constant c > 0, independent of
b, for which

    c ·1 + c_b^-2∇ + b u_2 + ∇ u_2≤√(ℒ^A_b)u_2≤c ·1 + c_b^2∇ + b u_2 + ∇ u_2

for all u ∈ H^1_b(^n).


To see that the above theorem is true, simply note that
(<ref>) implies both (H8Dα) and (H8Jα) for
α = 2 in this context. The independence of
the constant c from b follows from the fact that (H2) is satisfied
by Γ_J, B_1, B_2 with constant independent of b.










 



§ LITERATURE COMPARISON


 
It is important to note that this is not the first time that Kato-type
estimates have been studied for non-homogeneous operators. We will now
take some time to outline how our article differs in techniques and
results from each of these previous forays.

Recently, in <cit.> and
<cit.>, F. Gesztesy, S. Hofmann and R. Nichols
studied the domains of square root operators using techniques
distinct from those developed in <cit.>. The
article <cit.> considers potentials in the class
L^p + L^∞ but is not directly
relevant since it considers bounded
domains. On the other hand, <cit.> does not
impose a boundedness assumption on the domain and considers the
potential class L^n/2 + L^∞. There is already an immediate comparison with our
potential class since it was
shown in Proposition <ref> that L^n/2⊂𝒲 in dimension n > 4. It is not immediately clear whether
L^∞ is contained within our class.


Axelsson, Keith and McIntosh themselves considered non-homogeneous
operators on Lipschitz domains with mixed boundary conditions in
<cit.>. The potentials that they considered were, however,
bounded both from above and below and thus contained in RH_2⊂𝒲. In <cit.> and
<cit.>, M. Egert, R. Haller-Dintelmann and
P. Tolksdorf generalised this to certain non-smooth
domains. 

The articles <cit.>,
<cit.> and <cit.> are built upon the
original AKM framework, similar to this one.  A key step in the original proof of the
AKM framework is the proof of the estimate

    ∫^∞_0A_t - P_tu^2dt/t≲u^2.

This estimate allows for the A_t and P_t operators to be
freely interchanged at several stages in the proof granting use of
some of the more enviable properties of the A_t
operator. As has been demonstrated in this article through the
diagonalisation theorem, Theorem <ref>,
(<ref>) will not hold for a general potential. The
articles <cit.>, <cit.> and <cit.> circumvent this problem by imposing boundedness
of the potential from above and below. The boundedness of the
potential from above and below allows one to absorb the potential into
the multiplicative perturbation B_2 so that you are instead considering
the operators

    Γ_1 = [ 0 0 0; 1 0 0; ∇ 0 0 ],    B_1 = I    and    B_2 =
    [ I 0 0; 0 V 0; 0 0 A ].

For this set of operators, the large time-scale estimate

    ∫^∞_1Θ_t^B,1 P_t^1 u^2dt/t≲∫^∞_1P_t^1 u^2dt/t≲∫^∞_1t Π_1 P_t^1 u^2dt/t≲u^2

for u ∈ R (Γ_1) follows almost trivially. Then one only requires a small time-scale
version of  (<ref>) to
hold, namely

    ∫^1_0A_t - P_t^1u^2dt/t≲u^2

for all u ∈ R Γ_1. Such an
estimate is then proved to be true. 


This is a crude explanation
as to why the techniques developed in <cit.> cannot be
directly applied for a general potential that is not bounded both from
above and below.  There are similar obstructions, for example in
the selection of test functions in the Carleson measure
proof. However, these also disappear when the potential is bounded both
from above and below.

In this paper, our method has been to instead treat the potential as
part of the unperturbed operator and then to subsequently exploit the algebraic
structure of the operators Γ_V^1/2, B_1 and
B_2. This exploitation has allowed us to conclude that the estimate
 (<ref>) will at least hold on the third
 component which, it turns out, is all that is required. Similar obstructions in the proof of the main square
 function estimate also vanish when
 considered component-wise. As a consequence of this three-by-three mindset we have been able to obtain square
function estimates for potentials that aren't necessarily bounded from
above or below and, moreover, are not contained in
L^p^n for any 1 ≤ p ≤∞.

 








siam




 

 


