Higher Hölder regularity for a subquadratic nonlocal parabolic equation
Prashanta Garain Erik Lindgren Alireza Tavakoli


Extended high-ionization [Mgiv] emission tracing widespread shocks in starbursts seen by NIRSpec
    Miguel Pereira-Santaella<ref>
Ismael García-Bernete<ref>
Eduardo González-Alfonso<ref>
Almudena Alonso-Herrero<ref>
Luis Colina<ref>
Santiago García-Burillo<ref>
Dimitra Rigopoulou<ref>,<ref>
Santiago Arribas<ref>
Michele Perna<ref>

    April 29, 2024
=============================================================================================================================================================================================================================================




In this paper, we are concerned with the Hölder regularity for solutions of the nonlocal evolutionary equation

    ∂_t u+(-Δ_p)^s u = 0.
 
Here, (-Δ_p)^s is the fractional p-Laplacian, 0<s<1 and  1<p<2. We establish Hölder regularity with explicit Hölder exponents. We also include the inhomogeneous equation with a bounded inhomogeneity. In some cases, the obtained Hölder exponents are almost sharp. 
Our results complement the previous results for the superquadratic case when p≥ 2.







§ INTRODUCTION AND MAIN RESULT




In this article, we establish higher Hölder continuity of weak solutions for the subquadratic nonlocal parabolic equation

    ∂_t u+(-Δ_p)^s u=0.

Here 1<p<2 and 0<s<1. The operator (-Δ_p)^s is the fractional p-Laplace operator defined by

    (-Δ_p)^s u(x,t)=2 P.V.∫_ℝ^N|u(x,t)-u(y,t)|^p-2(u(x,t)-u(y,t))/|x-y|^N+ps dy,

where P.V. denotes the principal value.

The main contribution of this paper is to prove that solutions are almost Θ-regular in space and almost Γ-regular in time,
where 

    Θ(s,p):={[ s pp-1, s<p-1p,;        ;      1, s≥p-1p, ].    Γ(s,p):={[          1,     s<p-1p,;            ; 1s p-(p-2),     s≥p-1p. ].

We also allow a bounded right hand side in the equation, in which case these exponents are almost sharp. See Section <ref>.

These results are complementing the already existing Hölder estimates in <cit.>, valid for p≥ 2.



 §.§ Main result

Below we state our main theorem. For notation and relevant definitions, see Section <ref>.


Let Ω⊂ℝ^N be a bounded and open set, I=(t_0,t_1], 1<p< 2 and 0<s<1. Suppose (x_0,T_0)∈Ω× I is such that 

    Q_2R,(2R)^s p(x_0,T_0)⋐Ω× I.

Assume that u∈ L^p(J;W^s,p_loc(Ω))∩ L^p-1(J;L^p-1_sp(ℝ^N))∩ C(J;L^2_loc(Ω)) is a local weak solution of 

    ∂_t u+(-Δ_p)^s u=f    Ω× I,

as in Definition <ref> where f ∈ L^∞_(Ω× I)  and

    u_L^∞(Q_2R,(2R)^s p(x_0,T_0))+sup_-(2R)^sp+T_0<t≤ T_0Tail_p-1,s p(u(,t);x_0,R)   <∞.

Define the exponents

    Θ(s,p):={[ s pp-1, s<p-1p,;        ;      1, s≥p-1p, ].    Γ(s,p):={[          1,     s<p-1p,;            ; 1s p-(p-2),     s≥p-1p. ].

Then
    u∈ C^θ_x, loc(Ω× I)∩ C^γ_t, loc(Ω× I),    0<θ<Θ(s,p)   0<γ<Γ(s,p).
 
More precisely, for every (x_0,T_0) such that 

    Q_2R,(2R)^s p(x_0,T_0)⋐Ω× I,
 
there exists a constant C=C(N,s,p,θ,γ)>0 such that

    sup_t∈[T_0-(R/4)^s p,T_0] [u(·,t)]_C^θ(B_R/4(x_0))≤ CM^1+2-p/spR^-θ

and

    sup_x∈ B_R/4(x_0)[u(x,·)]_C^γ((-4^-spR^sp+T_0,T_0])≤ C MR^-γ,

where

    M=u_L^∞(Q_R,R^sp(x_0,T_0))+sup_t∈ (-R^sp+T_0,T_0]Tail_p-1,s p(u(·,t);x_0,R)^p-1 + R^spf_L^∞(Q_R,R^sp(x_0,T_0)) +1.





 §.§ Comments on our result

Regarding the almost sharpness of the spatial Hölder regularity, it suffices to study the stationary problem. Indeed, see in Example 1.5 in <cit.> and Section 1.2 in <cit.>, where it is proved that if the inhomogeneity is merely in L^q, then solutions may not be C^sp/(p-1) in general. See also Example 1.6 in <cit.>, where it is shown that for p=2 and s=1/2, solutions are not in general Lipschitz in space.

We point out that the time regularity is almost sharp in the case s p≤ (p-1). Indeed, in Remark 1.3 in <cit.> it is proved that solutions are not better than Lipschitz in time in general.

Finally, we wish to comment on the assumption on the tail. This is relevant, since there is a recent development on the regularity theory under the weaker assumption that the tail is merely in L^q for some q>p-1 (cf. <cit.>, <cit.> and <cit.>), while we in the present paper assume that the tail is in L^∞. This assumption is not unreasonable. Indeed, consider the following modified version of Example 5.2 in <cit.>.

Let 0<s<1 and 1<p<2 be such that sp≤ p-1. Suppose u_n is a weak solution of

    {[ ∂_tu_n + (-Δ_p)^su_n                    =                   0,         B_1× (-1,1],;                  u_n                    =                 g_n,  (ℝ^N∖ B_1)× (-1,1],;            u_n(·,-1)                    =                   0,                 B_1. ].

Here, g_n(x,t)=0 when x∈ℝ^N and t∈ (-1,0] while for x∈ℝ^N and t∈ (0,1],

    g_n(x,t)=δ f_n(t)+f_n'(t)χ_B_3∖ B_2(x),

where we  for n>e define

    f_n(t)=
    e^-1,   e^-1≤ t≤ 1,
    
     -tln t,   1/n<t<e^-1,
    
     tln n,    0<t≤1/n.

In addition, δ >0 is chosen so that 

    δ + (-Δ_p)^s χ_B_3∖ B_2(x)≤ 0,

for all x∈^N.  
The existence of such a solution u_n is guaranteed by for instance Theorem 1.4 in <cit.>. To this end, we note that since sp≤ (p-1)  <1, g_n lies in W^s,p(^N) and with this choice of f_n we have  f_n,f_n'∈ L^∞(-1,1]. Hence, g_n is admissible for this theorem. For x∈ B_1 and t∈ (-1,0], we have ∂_t g_n+(-Δ_p)^s g_n=0 since g_n(x,t)=0 for t∈ (-1,0]. We also see that f_n'(t)≥ 0 and therefore, for x∈ B_1 and t∈(0,1], we have

    ∂_t g_n+(-Δ_p)^s g_n = δ f_n'(t)+f_n'(t)(-Δ_p)^s χ_B_3∖ B_2(x)≤ 0.

Hence, g_n is a subsolution in B_1× (-1,1).  By the comparison principle (see for example Proposition A.3 in <cit.>), u_n≥ g_n for x∈ B_1 and t∈ (-1,1] and also u_n(x,t)=0 (note that g_n(x,t)=0 for x∈^N and t∈ (-1,0]) for all x∈ B_1 and t∈ (-1,0]. This implies that for x∈ B_1, 

    u_n(x,2/n)-u_n(x,-2/n)/2/n≥g_n(2/n)-g_n(0)/2/n=  g_n(2/n)/2/n=ln n-ln 2→∞ as n→∞.

Hence, the sequence {u_n} is not uniformly Lipschitz in time while 
    Tail_p-1,s p(u_n(·,t);0,1)^p-1≃ f_n^p-1(t)+(f_n')^p-1(t)
 is uniformly bounded in L^q(-1,1) for all q  <∞ but not for q=∞. This shows that we cannot have an apriori Lipschitz estimate in time, in terms of the L^q norm for any finite q. Hence, in some sense the L^∞-assumpiton on the tail is almost optimal.





and f'∈ L^q(-1,1) for any q <∞ but not for q=∞. Note that this implies that Tail_p-1,s p(u(·,t);0,1) is in L^q(-1,1) but not in L^∞(-1,1). Moreover, clearly f is not Lipschitz and therefore g is not Lipschitz in time. Now we choose δ >0 so that 

    δ + (-Δ_p)^s χ_B_3∖ B_2(x)≤ 0

for all x∈ B_1. Then

    g_t+(-Δ_p)^s g = δ f'(t)+f'(t)(-Δ_p)^s χ_B_3∖ B_2(x)+f”(t)χ_B_3∖ B_2(x)≤ 0

for x∈ B_1 and t∈ (-1,1), so that g is a subsolution in B_1× (-1,1). By comparison u≥ g. This implies that u cannot be Lipschitz in time.


By instead choosing f_n(t)∼ t^1- one can show for each q there is an upper bound on which Hölder seminorm (in time) can be bounded in terms of the L^q norm of the tail.



 §.§ Known results

The fractional p-Laplace equation

    (-Δ_p)^s u=0,   1<p<∞,

has been widely studied and the available literature is vast. For p=2, see <cit.> for a  local Hölder continuity result. We also refer to <cit.> and the references therein.

In the nonlinear case 1<p<∞, Hölder regularity and Harnack inequalities for weak solutions are proved in <cit.>. In <cit.>, Hölder continuity for viscosity solutions is established. Hölder continuity up to the boundary and fine boundary regularity has been studied in <cit.>

Higher regularity results for the nonlocal equation (<ref>) are established  in <cit.> for the case p≥ 2 and in <cit.> for p<2 respectively. 

Recently, higher Hölder regularity in the case p=2 with a more general kernel is established in <cit.>. We also mention <cit.>, where sharp regularity results for the equation with a right hand side belonging to a Lorentz space are studied.

The parabolic counterpart of the equation (<ref>), namely equation (<ref>), has recently been studied by various authors.

When p=2, local Hölder continuity of weak solutions for the equation (<ref>) is proved in <cit.>. A Harnack  inequality and local boundedness results can  for instance be found  in <cit.>. In addition, higher Sobolev regularity results can be found in <cit.>.

In the nonlinear parabolic case, when p>2, local boundedness result for the equation (<ref>) is proved in <cit.> with zero right-hand side. For the non-zero right-hand side, local boundedness results can be found in <cit.> for any 1<p<∞ and in <cit.> for any p≥ 2 respectively. Local Hölder continuity is proved in <cit.> for p>2 and in <cit.> for any 1<p<∞ respectively.

For higher Hölder regularity results, when p≥ 2, we refer to <cit.> with zero right hand side and <cit.> for non-zero right hand side respectively. Recently, higher Hölder regularity was obtained in the case p=2 allowing for more general kernel and a right hand side, in <cit.>.

It is worth mentioning that most of the results for nonlocal parabolic equations have been obtained under the assumption that the tail is locally bounded in time. Recently, there has been some advances when less restrictive assumptions on the tail have been assumed. This was initiated in <cit.> for the linear setting and considered for the case of the fractional p-Laplacian in <cit.> and <cit.>.

Plan of the paper: In Section 2, we introduce some notation and discuss preliminary results used throughout the paper. This is followed by Section 3, where we obtain an improvement of regularity on the Besov scale, given some initial Hölder regularity. This is then iterated and yields the desired spatial regularity in the normalized setting in Section 4. In Section 5, the time regularity is obtained, using the known spatial regularity. Finally, in Section 6, we prove our main result by combining the previous results.





§ PRELIMINARIES
 
In this section, we introduce some notation and present preliminary results needed in the rest of the paper. Throughout the paper, we shall use the following notation: B_r(x_0) denotes the ball of radius r with center at x_0. When x_0=0, we write B_r(0):=B_r.  Its Lebesgue measure
is given by 

    |B_r(x_0)|=ω(N)r^N,

where ω(N) is the volume of the unit ball B_1. We use the following notation for the parabolic cylinder

    Q_R,r(x_0,t_0)=B_R(x_0)×(t_0-r,t_0].

Again, when x_0=0 and t_0=0, we simply write Q_R,r.
The conjugate exponent l/l-1 of l>1 will be denoted by l'. We write
c or C to denote a positive constant which may vary from line to line or even in
the same line. The dependencies on parameters are written in the parentheses.

For any 1<q<∞, we define the monotone function J_q:ℝ→ℝ by

    J_q(t)=|t|^q-2t

and for 0<s<1 and 1<p<∞ we use the notation

    dμ= x  y/|x-y|^N+ps.


If ψ:ℝ^N→ℝ is a function and h∈ℝ^N, then  for any x∈ℝ^N, we define

    ψ_h(x)=ψ(x+h),    δ_h ψ(x)=ψ_h(x)-ψ(x)

and

    δ^2_h ψ(x)=δ_h(δ_h ψ(x))=ψ_2 h(x)+ψ(x)-2 ψ_h(x).

Let h∈ℝ^N and ϕ,ψ:ℝ^N→ℝ be two given functions, then the following discrete product rule holds:

    δ_h(ϕψ)(x)=(ψ_hδ_hϕ+ϕδ_hψ)(x),  ∀ x∈ℝ^N.

Similarly, for ϕ:ℝ^N× I→ℝ, for any fixed t∈ I and for every x∈ℝ^N, we define

    ϕ_h(x,t)=ϕ(x+h,t),    δ_h ϕ(x,t)=ϕ_h(x,t)-ϕ(x,t)

and

    δ^2_h ϕ(x,t)=δ_h(δ_h ϕ(x,t))=ϕ_2 h(x,t)+ϕ(x,t)-2 ϕ_h(x,t).

Function spaces: Let E⊂ℝ^N, N≥ 1 be an open and bounded subset.
Suppose v:E→ℝ is a function. Then for 0≤δ≤ 1, we define the δ-Hölder seminorm by

    [v]_C^δ(E):=sup_x≠ y∈ E|v(x)-v(y)|/|x-y|^δ.



It will be necessary to introduce two function spaces. For this reason, let 1≤ q<∞ and take  ψ∈ L^q(ℝ^N). For 0<β≤ 1, define

    [ψ]_𝒩^β,q_∞(ℝ^N):=sup_|h|>0δ_h ψ/|h|^β_L^q(ℝ^N).

The Besov-type spaces 𝒩^β,q_∞
are defined by 

    𝒩^β,q_∞(ℝ^N)={ψ∈ L^q(ℝ^N)  :  [ψ]_𝒩^β,q_∞(ℝ^N)<+∞},     0<β≤ 1.

The Sobolev-Slobodeckiĭ space is defined as

    W^β,q(ℝ^N)={ψ∈ L^q(ℝ^N)  :  [ψ]_W^β,q(ℝ^N)<+∞},     0<β<1,

where the seminorm [ · ]_W^β,q(ℝ^N) is given by

    [ψ]_W^β,q(ℝ^N)=(∬_ℝ^N×ℝ^N|ψ(x)-ψ(y)|^q/|x-y|^N+β q  x  y)^1/q.

The above spaces are endowed with the norms

    ψ_𝒩^β,q_∞(ℝ^N)=ψ_L^q(ℝ^N)+[ψ]_𝒩^β,q_∞(ℝ^N),
 
and

    ψ_W^β,q(ℝ^N)=ψ_L^q(ℝ^N)+[ψ]_W^β,q(ℝ^N).

We also introduce the space W^β,q(Ω) for a subset Ω⊂ℝ^N,

    W^β,q(Ω)={ψ∈ L^q(Ω)  :  [ψ]_W^β,q(Ω)<+∞},     0<β<1,

where naturally

    [ψ]_W^β,q(Ω)=(∬_Ω×Ω|ψ(x)-ψ(y)|^q/|x-y|^N+β q  x  y)^1/q.

The space W_0^β,q(Ω) is the subspace of W^β,q(ℝ^N) consisting of functions that are
identically zero in the complement of Ω.


Parabolic Banach spaces: Suppose that I⊂ℝ is an interval and let V be a separable and reflexive Banach space,
with a norm ·_V. Let V^* be its topological dual space. Assume that v is a function such v(t) belongs to V for almost every t∈ I. If the function t↦v(t)_V is measurable on I and 1≤ p≤∞, then v belongs to the Banach space L^p(I;V) if and only if

    ∫_Iv(t)_V^p dt<∞.

It is well known that the dual space of L^p(I;V) can be characterized by

    (L^p(I;V))^*=L^p(I;V^*).

See <cit.>. We also write v∈ C(I;V) if the function t↦ v(t) is continuous with respect to topology on V. Moreover, u is locally α-Hölder continuous (0<α<1) in space (respectively, locally β-Hölder continuous (0<β<1) in time) on Ω× I and write

    u∈ C^α_x,loc(Ω× I)   (respectively, u∈ C^β_t,loc(Ω× I))

if for any compact subset K× J⊂Ω× I, we have 

    sup_t∈ J[u(·,t)]_C^α(K)<∞  (respectively, sup_x∈ K[u(x,·)]_C^β(J)<∞).



Tail spaces and weak solutions:  The so-called tail spaces are expedient for nonlocal equations. The tail space is defined as

    L^q_α(ℝ^N)={u∈ L^q_ loc(ℝ^N)  : ∫_ℝ^N|u|^q/1+|x|^N+α  x<+∞},     q>0 α>0,

and is endowed with the norm

    u_L^q_α(ℝ^N)=(∫_ℝ^N|u(x)|^q/1+|x|^N+α  x)^1/q.

The  global behavior of a function u∈ L^q_α(ℝ^N) is measured by the quantity

    Tail_q,α(u;x_0,R)=[R^α ∫_ℝ^N∖ B_R(x_0)|u(x)|^q/|x-x_0|^N+α  x]^1/q.

Here x_0∈ℝ^N, R>0, β>0.

(Local weak solution)
Let 1<p≤ 2 and 0<s<1. Suppose Ω⊂ℝ^N is an open and bounded set and that f∈ L^∞_loc(Ω× I). For any t_0,t_1∈ℝ, we define I=(t_0,t_1]. We say that u is a local weak solution of the equation

    ∂_t u+(-Δ_p)^s u= f  in Ω× I

if for any closed interval J=[T_0,T_1]⊂ I, the function u is such that

    u∈ L^p(J;W^s,p_loc(Ω))∩ L^p-1(J;L^p-1_sp(ℝ^N))∩ C(J;L^2_loc(Ω))

and it satisfies

    -∫_J∫_Ωu(x,t)∂_t ϕ(x,t)  x  t+∫_J∫_ℝ^N∫_ℝ^NJ_p(u(x)-u(y))(ϕ(x)-ϕ(y)) μ  t
       =∫_Ω u(x,T_0)ϕ(x,T_0)   x-∫_Ω u(x,T_1)ϕ(x,T_1)   x    + ∫_J∫_Ω f(x,t) ϕ(x,t)  x  t ,

for every ϕ∈ L^p(J;W^s,p(Ω))∩ C^1(J;L^2(Ω)) which has spatial support compactly contained in Ω. Here 

    μ= x  y/|x-y|^N+ps

as defined in (<ref>).




We have the following embedding result from <cit.>.

Let ψ∈𝒩_∞^β,q(ℝ^N), where 0<β<1 and 1≤ q<∞ such that β q>N. Then for every 0<α<β-N/q, we have ψ∈ C^α_loc(ℝ^N). More precisely,

    sup_x≠ y|ψ(x)-ψ(y)|/|x-y|^α≤ C([ψ]_𝒩_∞^β,q(ℝ^N))^α q+N/β q(ψ_L^q(ℝ^N))^(β-α)q-N/β q,

for some positive constant C=C(N,q,α,β) which blows up as α↗β-N/q.


The result below allows for power functions of the solution to be used as test functions. It can be proved by following the proof of <cit.>, see also <cit.>.


Let a>0, 1<p≤ 2 and 0<s<1. Suppose that u is a local weak solution of 

    ∂_t u+(-Δ_p)^s u=f  in B_2× (-2^spa,0]

as in Definition <ref>,   where 

    f ∈ L_loc^∞(B_2 × (-2^spa,0] )   and   u∈ L^∞ (E× (-a,0]),

for every E⋐ B_2.
Let η be a non-negative Lipschitz function, with compact support in B_2. Let τ be a smooth non-negative function such that 0≤τ≤ 1 and

    τ(t)=0 for t≤ T_0,  τ(t)=1 for t≥ T_1

for some -a<T_0<T_1<0. Then, for any locally Lipschitz function F:ℝ→ℝ and any h∈ℝ^N such that 0<|h|<1/4(supp η,∂ B_2), we have

    ∫_T_0^T_1∬_ℝ^2N(J_p(u_h(x,t)-u_h(y,t))-J_p(u(x,t)-u(y,t)))
       ×(F(δ_h u(x,t))η(x)^2-F(δ_h u(y,t))η(y)^2)τ(t)  μ t
       +∫_B_2ℱ(δ_h u(x,T_1))η(x)^2  x
    =∫_T_0^T_1∫_B_2ℱ(δ_h u)η^2τ'  x  t   + ∫_T_0^T_1∫_B_2 (δ_h f)F(δ_h u) η^2 τ x  t ,

where ℱ(t)=∫_0^tF(ρ) dρ.


The proof follows exactly as in the proof of <cit.> as well as <cit.> by replacing η^p with η^2.




§ IMPROVED BESOV REGULARITY

 In this section, we obtain an improvement of regularity on the Besov scale, given some initial Hölder regularity.  

    Let 1<p<2, 0<s<1 and a>0. Suppose that u is a local weak solution of 

    ∂_t u + (-Δ_p)^s u= f   in   B_2× (-2^spa,0]
 
 where  
    
    f_L^∞ (B_1× (-a,0]) ≤ 1,   u_L^∞(B_1× (-a,0])≤ 1,

    
    
    sup_-a<t≤ 0Tail_p-1,s p(u(,t);0,1) ≤ 1  and  sup_-a<t ≤ 0[u(,t)]_C^γ(B_1)≤ 1,

    for some γ∈ [0,1). Assume also that for some α∈ [0,1), 1≤ q< ∞ and 0<h_0< 1/10, we have
    
    ∫_T_0^0sup_0 < h < h_0δ_h^2 u/h^α_L^q(B_R+ h_0)^q   t< ∞

     for R such that 4h_0 < R ≤ 1-5h_0, and for T_0 such that -a < T_0 < 0. Then for any ρ such that T_0 <ρ+ T_0<0, we have
    
    ∫_T_0 +ρ^0   sup_0<h<h_0δ_h^2 u/h^sp-γ(p-2) + α q/q+1_L^q+1(B_R-4h_0)^q+1 t +sup_T_0+ρ < t ≤ 0sup_0< h <h_0∫_B_R-4h_0δ_h u(x,t)^q + 1/h^α q x 
       ≤C/min{ρ, 1 }∫_T_0^0( 1+ sup_0<h<h_0δ_h^2 u/h^α_L^q(B_R+h_0)^q )  t,

    for some positive constant C=C(N,s,p,q,h_0,γ).


We prove the result in two steps. In the first, we prove 

    ∫_T_0+ ρ^T_1sup_0< h<h_0∫_B_R-4h_0   | δ_h^2 u/h^sp-γ(p-2) + α q/q +1|^q + 1 x  t + sup_0< h <h_0∫_B_R-4h_0δ_h u(x,T_1)^q + 1/h^α q x
       ≤C/min{ρ, 1 }∫_T_0^T_1( 1+ sup_0<h<h_0δ_h^2 u/h^α_L^q(B_R+h_0)^q )  t,

 with the upper limit T_1<0 and in Step 2, we prove (<ref>) for T_1=0, then (<ref>) follows by taking the supremum over T_1.

Step 1: Estimate for T_1<0.  Assume ρ+T_0<T_1<0, let r= R-4h_0 and take η to be a non-negative smooth function on ℝ^N such that
    
    0≤η≤ 1 on B_R,  η≡ 1   on   B_r,  η≡ 0 in ℝ^N∖ B_R+r/2  and  ∇η≤C/h_0 on B_R+r/2,

     for some positive constant C.
    
Take also  τ:→ to be a smooth function such that 0 ≤τ≤ 1 in ℝ and 

    τ≡ 1   on   [T_0+ρ , ∞),   τ≡ 0   on  (-∞ , T_0],   and  τ^'≤C/ρ in ℝ,

for some positive constant C. Let 1≤ q<∞, 0<|h|<h_0 and α∈[0,1) be as in the statement of the theorem and define ϑ=α-1/q. Lemma <ref>  with F(t)= J_q+1(t) implies upon dividing with h^1+ ϑ q that
    
    ∫_T_0^T_1   ∬_ℝ^N×ℝ^N(J_p(u_h(x,t)-u_h(y,t))-J_p(u(x,t)-u(y,t)))/|h|^1+ϑ q
       ×(J_q+1(δ_h u(x,t)) η(x)^2-J_q+1(δ_hu(y,t)) η(y)^2)τ(t) μ  t
       +1/q+1∫_B_2|δ_h u(x,T_1)|^q+1/|h|^1+ϑ q η^2  x 
       =1/q+1∫_T_0^T_1∫_B_2|δ_h u|^q+1/|h|^1+ϑ q η^2  τ'  x  t 
       + ∫_T_0^T_1∫_B_2δ_h f J_q+1(δ_h u)/h^1+ϑ qη^2 τ x  t.

The triple integral is now divided into three pieces:

    ℐ_i:=∫_T_0^T_1ℐ_i(t) τ(t)    dt,      i=1,2,3,

where we for t∈ (T_0,T_1) define

    _1(t):=∬_B_R× B_R   (J_p(u_h(x,t)-u_h(y,t))-J_p(u(x,t)-u(y,t)))/|h|^1+ϑ q
       ×(J_q+1(δ_h u(x,t)) η(x)^2-J_q+1(δ_h u(y,t)) η(y)^2) μ ,


    _2(t):=∬_(x,y)∈ B_R+r/2× ( ℝ^N∖ B_R)   (J_p(u_h(x,t)-u_h(y,t))-J_p(u(x,t)-u(y,t)))/|h|^1+ϑ q J_q+1(δ_h u(x,t)) η(x)^2 μ ,

and

    _3(t):=-∬_(x,y)∈ (ℝ^N∖ B_R)× B_R+r/2   (J_p(u_h(x,t)-u_h(y,t))-J_p(u(x,t)-u(y,t)))/|h|^1+ϑ q J_q+1(δ_h u(y,t)) η(y)^2 μ,

where we used that η vanishes identically outside B_R+r/2. We also denote the terms on the right-hand side of (<ref>) by _4 and _5:

    _4:= 1/q+1∫_T_0^T_1∫_B_2|δ_h u|^q+1/|h|^1+ϑ q η^2 τ'  x  t,

and

    _5:=  ∫_T_0^T_1∫_B_2δ_h f J_q+1(δ_h u)/h^1+ϑ qη^2 τ x  t.

Therefore, with the above notation, equation (<ref>) reduces to the following equation

    _1 +1/q+1∫_B_2|δ_h u(x,T_1)|^q+1/|h|^1+ϑ q η^2  x = -_2 -_3 + _4.

Estimate of ℐ_1: Since 1<p<2, by the same argument as in the proof of <cit.>, we obtain for any t∈(T_0,T_1)

    ℐ_1(t)
    ≥ c^-1[ δ_h u^q-1/2δ_h u/h^1+ϑ q/2η]_W^σ,2(B_R)^2 - C ∫_B_Rδ_h u(x,t)^p+q -1/h^1+ ϑ q - C ∫_B_Rδ_h u(x,t)^q +1/h^1+ ϑ q.

After integration with respect to t this becomes

    ℐ_1   ≥ c^-1∫_T_0^T_1[ δ_h u^q-1/2δ_h u/h^1+ϑ q/2η]_W^σ,2(B_R)^2τ dt - C ∫_T_0^T_1∫_B_Rδ_h u(x)^p+q -1/h^1+ ϑ qτ x dt - C ∫_T_0^T_1∫_B_Rδ_h u(x)^q +1/h^1+ ϑ qτ x dt,

where c=c(p,q) and C=C(N,s,p,q,h_0) are positive constants, and 2σ = sp -γ(p-2). We remark that the properties of η and the given condition sup_-a≤ t ≤ 0[u(,t)]_C^γ(B_1)≤ 1 are used to derive the above estimate of ℐ_1. Using (<ref>) in (<ref>), we arrive at

    ∫_T_0^T_1   [ δ_h u^q-1/2δ_h u/h^1+ϑ q/2η]_W^σ,2(B_R)^2 τ t  +∫_B_2|δ_h u(x,T_1)|^q+1/|h|^1+ϑ q η^2  x 
       ≤ C( ∫_T_0^T_1∫_B_Rδ_h u(x,t)^q +1/h^1+ ϑ qτ x  t + ∫_T_0^T_1∫_B_Rδ_h u(x,t)^p+q -1/h^1+ ϑ qτ x 
      t) 
       + c(_2+ _3 +_4 + _5),

 where C=C(N,s,p,q,h_0) is a positive constant. 

Estimate of the nonlocal terms _2 and _3: First, we estimate _2(t) for any t∈ (T_0,T_1). Since |u|≤ 1 in B_1× (-a,0], we have

    (J_p(u_h(x,t)-u_h(y,t))-J_p(u(x,t)-u(y,t)))J_q + 1(δ_h u(x,t))≤ C(p)(1+u_h(y,t)^p-1 + u(y,t)^p-1)δ_h u(x,t)^q,

where x ∈ B_R+r/2, y ∈^N ∖ B_R and 4h_0 < R ≤ 1-5h_0. Therefore, x-y≥ h_0 y and we get

    ∫_^N ∖ B_R1 + u(y,t)^p-1 + u_h(y,t)^p-1/x-y^N+sp y 
         ≤ C(1+∫_^N ∖ B_Ru(y,t)^p-1/y^N+sp y
    + ∫_^N ∖ B_Ru_h(y,t)^p-1/y^N+sp y),

for some positive constant  C=C(N,s,p,h_0).
Now, since 4h_0<R, R<1, |u|≤ 1 in B_1×(-a,0] and Tail_p-1, sp(u(·,t);0,1)≤ 1 in (-a,0], we have

    ∫_^N ∖ B_Ru(y,t)^p-1/y^N+sp y    ≤∫_^N ∖ B_1u(y,t)^p-1/y^N+sp y + R^-N-sp∫_B_1u(y,t)^p-1 y 
       ≤∫_^N ∖ B_1u(y,t)^p-1/y^N+sp y + Nω_N (4h_0)^-N-sp
       ≤ C,

where C= C(N,s,p,h_0). As for u_h, we have 

    ∫_^N ∖ B_Ru(y+h,t)^p-1/y^N+sp y    ≤∫_^N ∖ B_R(h)u(y,t)^p-1/y-h^N+sp y ≤ 
    (3/2)^N+sp∫_^N ∖ B_R(h)u(y,t)^p-1/y^N+sp y 
       ≤ (3/2)^N+sp[ ∫_^N ∖ B_1u(y,t)^p-1/y^N+sp y + (R-h_0)^-N-sp∫_B_1u(y,t)^p-1 y ] 
       ≤ C( Tail_p-1,s p(u(·,t);0,1)^p-1 +(3h_0)^-N-sp) 
       ≤ C,

for some positive constant C=C(N,s,p,h_0). To obtain the above estimate, we have used that B_R(h) ⊂ B_1, 4h_0<R≤ 1-5h_0 and 

    y-h/y = y/y-h/y≥y/y - h/y≥ 1- h_0/R-h_0≥2/3.
 Using (<ref>) and (<ref>) in (<ref>), we get

    ∫_^N ∖ B_R1 + u(y,t)^p-1 + u_h(y,t)^p-1/x-y^N+sp y    ≤ C(N,s,p,h_0).

Using (<ref>) and (<ref>) and recalling the expression of _2(t), we get for any t∈ (T_0,T_1) that

    _2(t)   ≤∬_(x,y)∈ B_R+r/2×(^N ∖ B_R)| J_p(u_h(x,t)-u_h(y,t))-J_p(u(x,t)-u(y,t))|/x-y^N+sp δ_hu(x,t)^q/|h|^1+ϑ qη(x)^2   y  x 
       ≤C(N,s,p,h_0)∫_B_R+r/2δ_h u^q/h^1+ ϑ qη^2  x.

Multiplying by τ and integrating the above estimate from T_0 to T_1, we arrive at

    _2   ≤∫_T_0^T_1_2τ t 
     ≤ C  (∫_T_0^T_1(  ∫_B_Rδ_h u^q/h^1+ ϑ qη^2  x ) τ t ) 
    ≤ C ∫_T_0^T_1∫_B_Rδ_h u^q/h^1+ ϑ q x  t,

for some positive constant C=C(N,s,p,h_0). Here we have used that 0≤τ≤ 1 on ℝ.
Similarly,

    _3   ≤ C ∫_T_0^T_1∫_B_Rδ_h u^q/h^1+ ϑ q x  t,

for some positive constant C=C(N,s,p,h_0).


Estimate of _4 and _5: Using the properties of η and τ we have

    _4 = 1/q +1|  ∫_T_0^T_1∫_B_2|δ_h u|^q+1/|h|^1+ϑ qη^2 τ^' x  t | ≤C/ρ∫_T_0^T_1∫_B_R+r/2|δ_h u|^q+1/|h|^1+ϑ q x  t,

for some positive constant C=C(N,p,q).
By the L^∞-bound on f together with the properties of η and τ, we also obtain

    _5=   | ∫_T_0^T_1∫_B_2δ_h f J_q+1(δ_h u)/h^1+ ϑ qη^2 τ x  t  | ≤∫_T_0^T_1∫_B_2δ_h fJ_q+1(δ_h u)/h^1+ ϑ qη^2 τ x  t 
       ≤δ_h f η_L^∞(B_2× (T_0,T_1))∫_T_0^T_1∫_B_2δ_h u^q/h^1+ ϑ qη x  t 
       ≤ 2∫_T_0^T_1∫_B_Rδ_h u^q/h^1+ ϑ q x  t.

Using (<ref>), (<ref>), (<ref>) and (<ref>) in (<ref>), we arrive at

    ∫_T_0 +ρ^T_1   [ δ_h u^q-1/2δ_h u/h^1+ϑ q/2η]_W^σ,2(B_R)^2  t  + c/q+1∫_B_R|δ_h u(x,T_1)|^q+1/|h|^1+ϑ q η^2  x 
       ≤ C( ∫_T_0^T_1∫_B_Rδ_h u(x,t)^q +1/h^1+ ϑ qτ x  t + ∫_T_0^T_1∫_B_Rδ_h u(x,t)^p+q -1/h^1+ ϑ qτ x 
      t) 
         + C( 1/ρ∫_T_0^T_1∫_B_Rδ_h u(x,t)^q +1/h^1+ ϑ q x  t + ∫_T_0^T_1∫_B_Rδ_h u(x,t)^q /h^1+ ϑ q x  t  ) 
       ≤C/min{ρ,1}∫_T_0^T_1∫_B_R(δ_h u(x,t)^q/h^1+ ϑ q + δ_h u(x,t)^q +1/h^1+ ϑ q +δ_h u(x,t)^p +q - 1/h^1+ ϑ q)  x  t 
       ≤C/min{ρ,1}∫_T_0^T_1∫_B_Rδ_h u(x,t)^q/h^1+ ϑ q x  t,

where C=C(N,s,p,q,h_0)>0 is a positive constant. Here we have used that |τ|≤ 1 and that |δ_h u(x,t)|≤ 2 for (x,t)∈ B_R× (-a,0]⊂ B_1× (-a,0]. Next, we will find a lower bound of the first integral of the left hand side of (<ref>), that is

    ∫_T_0 +ρ^T_1   [ δ_h u^q-1/2δ_h u/h^1+ϑ q/2η]_W^σ,2(B_R)^2  t.

Indeed, by following the same reasoning as in the proof of <cit.>, we obtain for any 0< ξ, h < h_0 that 

    ∫_B_r| δ_ξδ_h u/ξ^2σ/q+1h^1+ϑ q/q +1|^q +1 x    ≤  C [  δ_h u^q-1/2δ_h u/h^1+ϑ q/2η]_W^σ , 2(B_R)^2 + C ∫_B_Rδ_h u^q +1/h^1+ ϑ q x ,

where C= C(N,s,q,σ,h_0)>0 is a positive constant. We take ξ=h in (<ref>), take supremum over 0< h < h_0 and integrate over the time interval (T_0+ρ,T_1) to arrive at

    ∫_T_0+ρ^T_1sup_0<h<h_0 ∫_B_r| δ_h^2 u/h^2σ/q+1h^1+ϑ q/q +1|^q +1 x  t     ≤ C ∫_T_0+ ρ^T_1sup_0<h<h_0 [  δ_h u^q-1/2δ_h u/h^1+ϑ q/2η]_W^σ , 2(B_R)^2  t 
          + C ∫_T_0+ ρ^T_1sup_0<h<h_0 ∫_B_Rδ_h u^q +1/h^1+ ϑ q x  t,

where C=C(N,s,q,σ,h_0)>0  is a positive constant. By using (<ref>) in (<ref>) we obtain


    ∫_T_0+ρ^T_1   sup_0< h<h_0 ∫_B_r| δ_h^2 u/h^2σ/q+1+ 1+ϑ q/q +1|^q +1 x  t + sup_0< h <h_0∫_B_Rδ_h u(x,T_1)^q + 1/h^1+ ϑ qη^2  x 
       ≤C/min{ρ,1}∫_T_0^T_1sup_0< h<h_0∫_B_Rδ_h u(x)^q/h^1+ ϑ q x  t  + C ∫_T_0+ ρ^T_1sup_0<h<h_0 ∫_B_Rδ_h u^q +1/h^1+ ϑ q x  t  
       ≤C/min{ρ,1}∫_T_0^T_1sup_0< h<h_0∫_B_Rδ_h u(x)^q/h^1+ ϑ q x  t ,

where C depends on N,s,p,q,σ,  and h_0. In the last inequality, we have again used the fact that δ_h u(x,t)≤ 2 for x ∈ B_R and t ∈ (-a,0].

Recalling that ϑ= α - 1/q and 2 σ = sp-γ (p-2), we obtain from (<ref>) that

    ∫_T_0+ ρ^T_1sup_0< h<h_0∫_B_r   | δ_h^2 u/h^sp-γ(p-2) + α q/q +1|^q + 1 x  t + sup_0< h <h_0∫_B_rδ_h u(x,T_1)^q + 1/h^α q x
       ≤C/min{ρ,1}∫_T_0^T_1sup_0<h<h_0∫_B_Rδ_h u^q/h^α q x  t,

for some positive constant C=C(N,s,p,q,h_0,γ).

Since 1<q<∞ and |u|≤ 1 in B_1×(-a,0], we may for α∈ (0,1) use the second estimate of <cit.> to estimate the first order difference quotient in the right hand side of (<ref>) by a second order difference quotient. Then (<ref>) transforms into the
inequality below upon recalling the relations between R, r and h_0:

    ∫_T_0+ ρ^T_1sup_0< h<h_0∫_B_R-4h_0   | δ_h^2 u/h^sp-γ(p-2) + α q/q +1|^q + 1 x  t + sup_0< h <h_0∫_B_R-4h_0δ_h u(x,T_1)^q + 1/h^α q x
       ≤C/min{ρ, 1 }∫_T_0^T_1( 1+ sup_0<h<h_0δ_h^2 u/h^α_L^q(B_R+h_0)^q )  t,

for some positive constant C=C(N,s,p,q,h_0,γ). If α=0, we may directly use that |u|≤ 1  in  B_1×(-a,0] together with (<ref>) to arrive at (<ref>).





Step 2: Conclusion for T_1=0.
In this case, the previous proof does not directly work, since it relies on Lemma <ref>, which requires T_1<0. However, the constant C in (<ref>) does not depend on T_1, we can therefore use a limiting argument. By assumption, we have for some α∈[0,1), 1<q<∞ and 0<h_0<1/10, that

    ∫_T_0^0sup_0<|h|< h_0δ_h ^2 u/|h|^α^q_L^q(B_R+h_0)<∞

for 4h_0<R≤ 1-5h_0 and -a<T_0<0. For fixed ρ and T_0, (<ref>) implies that for every T_1<0 such that ρ+T_0<T_1, there holds

    ∫_T_0+ ρ^T_1sup_0< h<h_0∫_B_R-4h_0   | δ_h^2 u/h^sp-γ(p-2) + α q/q +1|^q + 1 x  t + sup_0< h <h_0∫_B_R-4h_0δ_h u(x,T_1)^q + 1/h^α q x
       ≤C/min{ρ, 1 }∫_T_0^T_1( 1+ sup_0<h<h_0δ_h^2 u/h^α_L^q(B_R+h_0)^q )  t,

for some positive constant C=C(N,s,p,q,h_0,γ).
By the monotone convergence theorem,

    lim_T→ 0^-∫_T_0+ ρ^Tsup_0< h<h_0∫_B_R-4h_0| δ_h^2 u/h^sp-γ(p-2) + α q/q +1|^q + 1 x  t 
       =∫_T_0+ ρ^0sup_0< h<h_0∫_B_R-4h_0| δ_h^2 u/h^sp-γ(p-2) + α q/q +1|^q + 1 x  t.

In addition, by the definition of local weak solution

    t↦δ_h u(·,t)/|h|^α q/q+1

is  locally a  continuous function on (-2^spa,0] with values in L^2(B_r) for every fixed 0<|h|<h_0. Therefore, 

    lim_T→ 0^-δ_h u(·,T)/|h|^α q/q+1-δ_h u(·,0)/|h|^α q/q+1_L^2(B_R-4h_0)=0.

Since q≥ 1, this in turn implies that[We use the following standard fact: if {f_n}_n∈ℕ converges to f in L^m(E), then

    lim_n→∞inff_n_L^k(E)≥f_L^k(E)

for any  k ≥ m.]

    liminf_T→ 0^-δ_h u(·,T)/|h|^α q/q+1_L^q+1(B_R-4h_0)≥δ_h u(·,0)/|h|^α q/q+1_L^q+1(B_R-4h_0)

for every 0<|h|<h_0. Combining (<ref>) and (<ref>) with (<ref>), gives  that  estimate (<ref>) holds also for T_1=0. Finally, by taking the supremum over T_1∈ [T_0+ρ,0] we obtain the desired estimate (<ref>).








§ SPATIAL REGULARITY

Using Proposition <ref>, we establish the following spatial regularity results.


    Assume that a≥ a_0>0, 0<κ<1, 1<p<2, 0<s<1, and γ∈ [0,1). Suppose u is a local weak solution of 

    ∂_t u + (-Δ_p)^s u= f     B_2× (-2^spa,0]
 
as in Definition <ref>,  where 
    
    f_L^∞ (B_1× (-a,0]) ≤ 1,   u_L^∞(B_1× (-a,0])≤ 1,

    
    sup_-a<t≤ 0Tail_p-1,s p(u(·,t);0,1) ≤ 1,  and  sup_-a< t ≤ 0[u(·,t)]_C^γ(B_1)≤ 1,

    for some γ∈ [0,1). Let τ = min{ sp-γ(p-2) , 1}.  Then  for any ϵ∈ (0, τ) we have 
    
    sup_-a(1-κ) ≤ t ≤ 0[u(·, t)]_C^τ -ϵ (B_1/2)≤ C,

    for some positive constant C=C(N,s,p,ϵ,γ, κ a_0).


    Take 0<ϵ < τ and choose q_0=q_0(N,) so that
    
    τ -ϵ/2 - N/q_0 > τ - ϵ > 0   and  q_0/q_0+1(τ-ϵ/4) ≥τ -ϵ/2 .

    Then consider the sequence
    
    α_0 = 0 ,   α_i+1:= sp-γ(p-2) + α_i q_0/q_0 +1,    i= 0, … , i_∞,

where we choose i_∞=i_∞(N,p,s,,γ)∈ℕ such that 
    
    α_i_∞ -1 < τ -ϵ/4≤α_i_∞ .

    This is possible since the sequence of α_i is increasing and 
    
    lim_i →∞α_i = sp- γ (p-2).
 
    Define also
    
    h_0= 1/40 i_∞,     R_i = 7/8 - (5i+1)h_0,   for  i= 0, … , i_∞ .

    Note that
    
    R_0 + h_0 = 7/8  and   R_i_∞ -1 - 4h_0 = 3/4.

 Now we apply Proposition <ref> (ignoring the second term in the left-hand side of (<ref>)) and apply Young's inequality with
    
    q=q_0,   ρ = κ a/2(i_∞ +1),    T_0 = -a+ (i+1)ρ,   R=R_i.

 We observe that 0<ρ<-T_0 for the above choice of T_0, since κ∈(0,1). Further, R_i-4h_0 = R_i+1 +h_0. Therefore, we obtain the following iterative scheme of inequalities:

    ∙ For i=0, we get
    
    ∫_-a + 2 ρ^0sup_0<h<h_0δ_h^2 u/h^sp-γ(p-2)  /q_0_L^q_0(B_R_1 + h_0)^q_0 t 
        ≤C/min{ρ, 1 }∫_-a+ρ^0( sup_0<h< h_0δ_h^2 u_L^q_0(B_7/8)^q_0 +1 )  t.

Notice that sp-γ(p-2)/q_0+1 = α_1.


    ∙ For i=1, … i_∞-2, we have
    
    ∫_-a + (i+2)ρ^0sup_0<h< h_0δ_h^2 u/h^α_i+1_L^q_0(B_R_i+1 + h_0)^q_0 t 
        ≤C/min{ρ, 1 }∫_-a + (i+1)ρ^0( sup_0<h< h_0δ_h^2 u/h^α_i_L^q_0(B_R_i + h_0)^q_0 +1)  t.


    ∙ For i= i_∞-1, we have
    
    ∫_-a + κ a/2^0sup_0<h< h_0δ_h^2 u/h^α_i_∞_L^q_0(B_3/4)^q_0 t ≤C/min{ρ, 1 }∫_-a + i_∞ρ^0( sup_0<h< h_0δ_h^2 u/h^α_i_∞-1_L^q_0(B_R_i_∞-1 + h_0)^q_0 +1) t.

    Here C= C(N,s,p,ϵ,γ,h_0). Also, since u_L^∞(B_1×(-a,0])≤ 1, we have
    
    sup_0< h< h_0δ_h^2 u_L^q_0(B_7/8)≤ 3 .

    Hence, using (<ref>) in the above iterative scheme of inequalities (<ref>), (<ref>) and (<ref>), we have the following estimate 
    
    ∫_-a(1- κ/2) ^0sup_0<h< h_0δ_h^2 u/h^α_i_∞_L^q_0(B_3/4)^q_0 t ≤C(N,s,p,ϵ,γ,h_0)/(min{κ a,1})^i_∞.

    As τ -ϵ/4≤α_i_∞, for all h≤ 1 we have 
    
    h^α_i_∞≤h^τ - ϵ/4.

    Using this in (<ref>) we arrive at
    
    ∫_-a(1- κ/2) ^0sup_0<h< h_0δ_h^2 u/h^τ -ϵ/4_L^q_0(B_3/4)^q_0 t ≤C(N,s,p,ϵ,γ,h_0)/(min{κ a,1})^i_∞.

Now we set
    
    T_1=0,    T_0 = -a(1-κ/2),   ρ= κ a/2,   α = τ - ϵ/4,  and   q=q_0.

    Then,
    
    R+h_0=3/4,    R- 4h_0 = 3/4-5h_0 ≥5/8.

    Since a>0 and κ∈(0,1), we have 0<ρ<T_1-T_0. We now apply Proposition <ref> again, taking (<ref>) into account (ignoring the first term in the left-hand side of (<ref>) this time). This yields
    
    sup_-a(1-κ)≤ t≤ 0 sup_0< h <h_0δ_h u(,t)/h^(τ - ϵ/4) q_0/q_0 +1_L^q_0(B_5/8)^q_0   ≤C/min{κ a,1}∫_-a(1-κ a/2)^0( sup_0<h< h_0δ_h^2 u( , t)/h^τ - ϵ/4_L^q_0(B_3/4)^q_0 +1 )  t 
       ≤C/(min{κ a,1})^i_∞ +1,

for some positive constant C=C(N,s,p,ϵ,γ,h_0). Here we have again used Young's inequality to reduce the power to q_0.
    Since q_0/q_0+1≥τ -ϵ/2/τ - ϵ/4, we have
    
    (τ -ϵ/4) q_0/ q_0 +1 ≥τ -ϵ/2.

    By using this in (<ref>) we arrive at
    
    sup_-a(1-κ) ≤ t≤ 0    sup_0< h <h_0δ_h u(,t)/h^τ - ϵ/2_L^q_0(B_5/8)
       ≤sup_-a(1-κ)≤ t≤ 0 sup_0< h <h_0δ_h u(,t)/h^(τ - ϵ/4) q_0/q_0 +1_L^q_0(B_5/8)≤C(N,ϵ,p,s,γ,h_0)/(min{κ a,1})^i_∞ +1.

    Now we take χ∈ C^∞_c(B_9/16) such that
    
    0≤χ≤ 1 in B_9/16,  χ≡ 1  in  B_1/2,   ∇χ≤ C in B_9/16,

  for some positive constant C. 
    In particular, for any h with 0<h< h_0, we have
    
    δ_h χ/h^τ - ϵ/2≤δ_h χ/h≤ C.

    Also, recall that 
    
    δ_h( χ u) = χ_h δ_h u + u δ_h χ .

    Hence, using the above properties of χ, 
 for every t ∈ [-a(1-κ),0],  we have
    
    [uχ]_^τ -ϵ/2,q_0_∞(^N)= sup_h>0δ_h (uχ)/h^τ - ϵ/2_L^q_0(^N)
       ≤sup_ 0< h< h_0δ_h (uχ)/h^τ - ϵ/2_L^q_0(^N)  + sup_h≥ h_0δ_h (uχ)/h^τ - ϵ/2_L^q_0(^N)
       ≤sup_ 0< h< h_0χ_h δ_h u/h^τ - ϵ/2_L^q_0(^N)  +   sup_ 0< h< h_0u δ_h χ/h^τ - ϵ/2_L^q_0(^N)
              + 1/h_0^τ - ϵ/2sup_h≥ h_0δ_h(u χ)_L^q_0(^N)
       ≤sup_ 0< h< h_0δ_h u/h^τ - ϵ/2_L^q_0(B_9/16 + h_0)  + u_L^q_0(B_9/16 + h_0)sup_0 < h < h_0δ_h χ/h^τ - ϵ/2_L^∞(B_9/16 +h_0)
            + 1/h_0^τ - ϵ/2sup_h≥ h_0( (u χ)_h_L^q_0(B_9/16(-h)) + u χ_L^q_0(B_9/16)) 
       ≤sup_ 0< h< h_0δ_h u/h^τ - ϵ/2_L^q_0(B_5/8)  + Cu_L^q_0(B_1),

    for some positive constant C=C(N,s,p,ϵ,γ,h_0). Since h_0 depends on s,p,ϵ,γ and u_L^∞(B_1×(-a,0])≤ 1, the above estimate combined with (<ref>) implies 
    
    sup_a(1-κ) ≤ t ≤ 0 [uχ(·,t)]_N^τ -ϵ/2,q_0_∞(^N)≤C(N,s,p,ϵ,γ)/(min{κ a,1})^c(N,s,p,,γ).

    Notice that by the choice of q_0, we have
    
    τ - ϵ < τ - ϵ/2 - N/q_0.
    
    Now we apply Theorem <ref> with q=q_0, β= τ - ϵ/2 and α = τ - ϵ to obtain 
    
    sup_-a(1-κ)≤ t ≤ 0 [u(·,t)]_C^τ - ϵ(B_1/2) = sup_-a(1-κ) ≤ t ≤ 0 [(u χ)( ,t))]_C^τ - ϵ(B_1/2)≤sup_-a(1-κ) ≤ t ≤ 0 [(uχ)(· ,t)]_C^τ - ϵ(^N)
       ≤ C sup_-a(1-κ) ≤ t ≤ 0( ( [uχ(·,t)]_^τ - ϵ/2, q_0_∞(^N))^(τ-ϵ)q_0+ N/(τ - ϵ/2)(q_0)(u χ(·,t)_L^q_0(^N))^(q_0)ϵ/2-N/(τ -ϵ/2)q_0) 
       ≤ C ( sup_-a(1-κ) ≤ t ≤ 0 [uχ(·,t)]_^τ - ϵ/2, q_0_∞(^N))^(τ-ϵ)q_0+ N/(τ - ϵ/2)(q_0)(sup_-a(1-κ) ≤ t ≤ 0u χ(·,t)_L^q_0(^N))^ϵ q_0/2-N/(τ -ϵ/2)q_0
       ≤C/(min{κ a,1})^c(τ-ϵ)q_0+ N/(τ - ϵ/2)q_0,

     where we to obtain the above estimate, have also used the estimate (<ref>) and the fact that u_L^∞(B_1× (-a,0])≤ 1.  Here C=C(N,s,p,ϵ,γ) and c=c(N,s,p,,γ) are positive constants. The proof follows by using that min{κ a,1}≥min{κ a_0,1} since a≥ a_0.



By yet another iteration process, we arrive at the final almost sp/(p-1)-regularity in space.



Let 1<p<2 and 0<s<1. Suppose u is a local weak solution of 

    ∂_t u+(-Δ_p)^s u= f     B_2× (-2^sp,0]
 
 where 

    u_L^∞(B_1× (-1,0])≤ 1,     f_L^∞(B_1 × (-1,0]))≤ 1 , 
           sup_t∈ (-1,0]Tail_p-1,s p(u(·,t);0,1)^p-1≤ 1.

Then for any ∈(0,Θ), there is  σ(,s,p)∈(0,1/2], such that u(·,t)∈ C^Θ-(B_σ) for all t∈ (-σ^sp,0], where

    Θ = min(sp/(p-1),1).

Moreover, 

    sup_t∈ (-σ^sp,0][u(·,t)]_C^Θ-(B_σ)≤ C(s,p,,N).


 The idea is to apply Proposition <ref> iteratively. Take ∈(0,Θ) and define

    γ_0=0,    γ_i+1=sp-γ_i(p-2)-(p-1)/2.

Then {γ_i} is an increasing sequence and γ_i→ sp/(p-1)-/2, as i→∞.  It is clear that there is i_∞=i_∞(s,p,)∈ℕ such that γ_i_∞≥Θ- and γ_i_∞-1<1.

Define 

    v_i(x,t)=u(2^-i x,2^-ispM_i^2-p t)/M_i

and

    M_i    = 1 + u_L^∞(B_2^-i× (-2^-i sp,0])+ sup_t∈ (-2^-isp,0]Tail_p-1,s p(u(·,t);0,2^-i) 
            +  2^-isp/(p-1)f_L^∞(B_2^-i×( -2^-isp ,0])^1/p-1 + 2^-iγ_isup_t∈ (2^-i sp,0][u(·,t)]_C^γ_i(B_2^-i)
       ≤ C(N,p,s,)(1+sup_t∈ (-2^-i sp,0][u(·,t)]_C^γ_i(B_2^-i)))
       =C(N,p,s,)(1+M_i-12^(i-1)γ_isup_t∈ (-M_i-1^p-22^-i sp,0][v_i-1(·,t)]_C^γ_i(B_1/2))
       ≤ C(N,p,s,)(1+M_i-1sup_t∈ (-M_i-1^p-22^-i sp,0][v_i-1(·,t)]_C^γ_i(B_1/2)).

Then v_i is a local weak solution of the equation v_t+(-Δ_p)^sv= f_i in B_2×(-2^sp M_i^p-2,0], where


    f_i(x,t)=2^-ispf(2^-ix,2^-ispM_i^2-pt)/M_i^p-1.

Moreover, if  

    sup_t∈ (-2^-isp,0][u(·,t)]_C^γ_i(B_2^-i)<∞,

then M_i <∞ and v_i and f_i are well defined and satisfy

    v_i_L^∞(B_1× (-M_i^p-2,0])≤ 1,    sup_t∈ (-M_i^p-2,0]Tail_p-1,s p(v_i(·,t);0,1)^p-1≤ 1,


    sup_t∈ (-M_i^p-2,0][v_i(·,t)]_C^γ_i(B_1)≤ 1,
       and   f_i_L^∞(B_1× (-M_i^p-2,0])≤ 1.

Now we apply Proposition <ref> with  a=M_i^p-2 and κ = 1-2^-sp to v_i in the cylinders B_2× (-2^spM_i^p-2,0] successively with γ=γ_i and  replaced by (p-1)/2 and obtain for i=1 to i=i_∞

    sup_t∈ (-2^-sp,0][v_0(·,t)]_C^γ_1(B_1/2)≤ C(s,p,,N), 
    
     M_1≤ C(s,p,,N)(1+ sup_t∈ (-2^-sp,0][v_0(·,t)]_C^γ_1(B_1/2))≤ C(s,p,,N),
    sup_t∈ (-M_1^p-22^-sp,0][v_1(·,t)]_C^γ_2(B_1/2)≤ C(s,p,,N),
    
     M_2≤ C(s,p,,N)(1+M_1 sup_t∈ (-M_1^p-22^-sp,0][v_1(·,t)]_C^γ_2(B_1/2))≤ C(s,p,,N),
    sup_t∈ (-M_i-1^p-22^-sp,0][v_i-1(·,t)]_C^γ_i(B_1/2)≤ C(s,p,,N),
    
     M_i≤ C(N,p,s,)(1+M_i-1sup_t∈ (-M_i-1^p-22^-i sp,0][v_i-1(·,t)]_C^γ_i(B_1/2))≤ C(s,p,,N),
    ⋯
    sup_t∈ (-M_i_∞-1^p-22^-sp,0] [v_i_∞-1(·,t)]_C^min(γ_i_∞,1-(p-1)/2)(B_1/2)≤ C(s,p,,N).

Note that at each step above we have used Proposition <ref> for a time interval with a=M_i^p-2≥ C(s,p,,N)=a_0 and we have also used (<ref>) to estimate M_i. If sp>1, we only do one iteration and in particular i_∞=1. In particular in that case (<ref>) becomes

    sup_t ∈ (-2^-sp,0] [v_0(,t)]_C^sp - ϵ(p-1)/2≤ C(s,p,ϵ,N).

Scaling back to u and using that γ_i_∞≥Θ-, we obtain 

    sup_t∈ (-2^-(i_∞)sp,0][u(·,t)]_C^Θ-(B_2^-i_∞)   =sup_t∈ (-M_i_∞-1^p-22^-sp,0]2^(i_∞-1)(Θ-)M_i_∞-1[v_i_∞-1(·,t)]_C^Θ-(B_1/2)
       ≤ C(s,p,,N).

This is the desired result with σ = 2^-i_∞.



one has to do a covering argument (sliding the center of the balls) in space variable to arrive at the following version:

Let 1<p<2 and 0<s<1. Suppose u is a local weak solution of 

    ∂_t u + (-Δ_p)^s u=0  in   B_2 × (-2^sp,0]
 
or rather in B_2 × (-2,0] if you like such that

    u_L^∞((-1,0]× B_1)≤ 1,   sup_-1<t≤ 0Tail_p-1,s p(u(,t);0,1) ≤ 1.

Then for any δ < Θ(s,p) we have

    sup_t∈ [-1/2^sp,0][u(,t)]_C^δ(B_1/2)≤ C(N,s,p,δ).







Now we apply Theorem <ref> to prove the following spatial regularity result.



Let Ω⊂ℝ^N be a bounded and open set, I=(t_0,t_1], 1<p<2 and 0<s<1. Suppose u is a local weak solution of 

    ∂_t u+(-Δ_p)^s u= f     Ω× I,

with f ∈ L^∞_(Ω× I) such that 

    u_L^∞(Q_2R,(2R)^s p(x_0,T_0))+sup_-(2R)^sp+T_0<t≤ T_0Tail_p-1,s p(u(,t);x_0,R)   <∞,

where (x_0,T_0) are such that 

    Q_2R,(2R)^s p(x_0,T_0)⋐Ω× I.
 
Then u∈ C^Θ-_x, loc(Ω× I) for every ∈(0,Θ), where Θ = min(sp/(p-1),1). 
More precisely, for every ∈(0,Θ), R>0 and every (x_0,T_0) such that 

    Q_2R,(2R)^s p(x_0,T_0)⋐Ω× I,
 
there  exist constants  C=C(N,s,p,)>0 and 0<σ(ϵ,s,p)≤1/2 such that

    sup_t∈(T_0-(σ R)^s p,T_0] [u(·,t)]_C^Θ-(B_σ R/2(x_0))≤C^1+2-p/spR^-(Θ-),

where

    = u_L^∞(Q_R,R^sp(x_0,T_0))+ sup_t∈ (T_0-R^sp,T_0]Tail_p-1,s p(u(·,t);x_0,R)^p-1 + R^spf_L^∞(Q_R,R^sp(x_0,T_0)) +1.



We perform the proof in the case x_0=0 and T_0=0.

Let

    u_R(x,t):=1/M u(RM^p-2/sp x+y_0,R^spt),    x∈ B_2,  t∈(-2^sp,0]

where M=M(R) is given by

    M=   (2+(2 N ω(N)/sp)^1/p-1) u_L^∞(B_R× (-R^sp,0])+ R^spf_L^∞(B_R ×(-R^sp,0])
        +2(1-σ)^-N-spsup_t∈ (-R^sp,0]Tail_p-1,s p(u(·,t);0,R)^p-1+1

and y_0 is chosen so that 

    σ RM^p-2/sp+|y_0|≤σ R

where  σ∈(0,1/2]  is as in Theorem <ref>. Since 1<p<2 and M≥ 1 
, (<ref>) implies

    2RM^p-2/sp+|y_0| ≤ 2RM^p-2/sp + σ R - σ R M^p-2/sp=RM^p-2/sp(2-σ)+σ R ≤ 2R.

Therefore u_R is a local weak solution of ∂_t u+(-Δ_p)^s u= f̃ in B_2× (-2^sp,0], where

    f̃(x,t)=R^sp/M f(RM^p-2/spx +y_0,R^spt) .


It is also straight forward to verify that (<ref>) implies

    u_R_L^∞(B_1× (-1,0])=M^-1u_L^∞(B_RM^p-2/sp(y_0)× (-R^sp,0])≤ M^-1u_L^∞(B_R× (-R^sp,0])≤ 1,

as well as

    f̃_B_1×(-1,0] = M^-1R^spf_L^∞(B_RM^p-2/sp(y_0)× (-R^sp,0])≤ M^-1R^spf_L^∞(B_R×(-R^sp,0])≤ 1.


We will now verify that also 

    sup_t∈ (-1,0]∫_ℝ^N∖ B_1|u_R(z,t)|^p-1/|z|^N+s p  z≤ 1.

We have

    ∫_ℝ^N∖ B_1|u_R(z,t)|^p-1/|z|^N+s p   dz=R^spM^(p-2)-(p-1)∫_^N∖ B_RM^p-2/sp(y_0)|u(y,R^spt)|^p-1/|y-y_0|^N+s p  y
       =R^spM^-1(∫_B_R ∖ B_RM^p-2/sp(y_0)|u(y,R^spt)|^p-1/|y-y_0|^N+s p  y  +∫_^N∖ B_R|u(y,R^spt)|^p-1/|y-y_0|^N+s p  y  )
       =I_1+I_2.



We note that |y_0|≤σ R, which implies that if y∈ B_R^c, then |y-y_0|≥ (1-σ) |y|. Therefore 

    sup_t∈ (-1,0]I_2≤ R^spM^-1(1-σ)^-N-spsup_t∈ (-R^sp,0]∫_^N∖ B_R|u(y,t)|^p-1/|y|^N+s p  y≤1/2
 
by the choice of M. For I_1, we instead have

    sup_t∈ (-1,0]I_1   ≤u^p-1_L^∞(B_R× (-R^sp,0])R^spM^-1∫_^N ∖ B_RM^p-2/sp1/|ỹ|^N+s p   dỹ
       =u^p-1_L^∞(B_R× (-R^sp,0])R^spM^-1∫^∞_RM^p-2/sp Nω(N) r^-1-sp   dr
       =u^p-1_L^∞(B_R× (-R^sp,0])M^-(p-1) Nω(N)/sp
       ≤1/2,

again by the choice of M.
By Theorem <ref>, u_R satisfies the estimate

    sup_t∈ (-σ^sp,0][u_R(x,t)]_C^Θ-(B_σ)≤ C,   C=C(N,s,p,).

By scaling back, we obtain

    sup_t∈ (-(σ R)^sp,0][u(x,t)]_C^Θ-(B_σ RM^p-2/sp(y_0) )≤ CM^1+2-p/sp (Θ-)R^-(Θ-).

Since this is valid for all y_0 satisfying (<ref>), we can, by varying y_0, cover the whole B_σ R/2 and obtain

    sup_t∈ (-(σ R)^sp,0][u(x,t)]_C^Θ-(B_σ R/2)≤ CM^1+2-p/spR^-(Θ-).

Once this is settled, this proves the theorem.

Let us provide the details of how to vary y_0. First of all, if M^p-2/sp≥ 1/2, then (<ref>) with y_0=0 implies (<ref>) directly. Therefore, we assume M^p-2/sp<1/2 in what follows.
We need to estimate 

    |u(x,t)-u(y,t)|/|x-y|^Θ-,   x,y∈ B_σ R/2,t∈ (-(σ R)^sp,0].

Take x,y∈ B_σ R/2 and fix t∈ (-(σ R)^sp,0]. If |x-y|≤σ RM^p-2/sp, it follows that 
    x,y∈ B_σ RM^p-2/sp((x-y)/2)
 and 

    σ RM^p-2/sp+|(x-y)/2|≤σ R/2+σ R/4≤  3σ R/4≤σ R

so that the choice y_0=(x-y)/2 is admissible for (<ref>). We may therefore apply (<ref>) directly by choosing y_0=(x-y)/2 to obtain 

    |u(x,t)-u(y,t)|/|x-y|^Θ-≤ CM^1+2-p/sp (Θ-)R^-(Θ-)≤ C M^1+ 2-p/sp R^-(Θ-).

 If instead |x-y|≥σ  RM^p-2/sp, we take
 
    z_i=x+y-x/⌈ M^2-p/sp⌉i,    i=0, …,  J=⌈ M^2-p/sp⌉.

Since x,y∈ B_σ R/2, we have |z_i-z_i+1| = x-y/⌈ M^2-p/sp⌉ < σ RM^p-2/sp, z_0=x and z_J=y. This choice implies that 
    z_i,z_i+1∈ B_σ RM^p-2/sp/2((z_i-z_i+1)/2)
 and

    σ RM^p-2/sp+|(z_i-z_i+1)/2|≤σ R/2+σ R/4 ≤σ R,

as before. Therefore, the choice y_0=(z_i-z_i+1)/2 is admissible for (<ref>). We can therefore apply (<ref>)  with y_0=(z_i-z_i+1)/2 and obtain 

    |u(z_i,t)-u(z_i+1,t)|/|z_i-z_i+1|^Θ-≤ CM^1+2-p/sp (Θ-)R^-(Θ-).

By the triangle inequality, we can now conclude

    |u(x,t)-u(y,t)|≤∑_i=0^J|u(z_i,t)-u(z_i+1,t)|   ≤∑_i=0^JCM^1+2-p/sp (Θ-)R^-(Θ-)|z_i-z_i+1|^Θ-
       ≤∑_i=0^JCM^1+2-p/sp (Θ-)R^-(Θ-)(x-y/⌈ M^2-p/sp⌉)^Θ-
       ≤ C J M^1+2-p/sp (Θ-)R^-(Θ -ϵ) M^p-2/sp(Θ -)x-y^Θ -
       ≤ CM^1+2-p/spR^-(Θ-)|x-y|^Θ -

where we have used that J= ⌈ M^2-p/sp⌉≤ M^2-p/sp+1≤ 2M^2-p/sp.
This proves (<ref>) and the proof is complete.

 We note that by a covering argument, we may, under the assumptions of Theorem <ref>, obtain the estimate

    sup_t∈(T_0-(R/2)^s p,T_0] [u(·,t)]_C^Θ-(B_R/2(x_0))≤ C^1+2-p/spR^-(Θ-),

where

    = u_L^∞(Q_R,R^sp(x_0,T_0))+ sup_t∈ (T_0-R^sp,T_0]Tail_p-1,s p(u(·,t);x_0,R)^p-1 + R^spf_L^∞(Q_R,R^sp(x_0,T_0)) +1.

Indeed, we may cover B_R/2(x_0)× (-(R/2)^sp+T_0,T_0] with a finite number of cylinders of the form 

    B_σ R/2(x_i)×(-(σ R/2)^sp+t_j,t_j],   i=1,…,n,   j=1,…,m,

where B_R/2(x_i)⊂ B_R(x_0) and (-(R/2)^sp+t_j,t_j]⊂ (-R^sp+T_0,T_0]. We may then apply Theorem <ref> to each of these cylinders and obtain 

    sup_t∈[t_j-(σ R/2)^s p,t_j] [u(·,t)]_C^Θ-(B_σ R/2(x_i))≤ CM^1+2-p/spR^-(Θ-),

where

    = u_L^∞(Q_R,R^sp(x_0,T_0))+ sup_t∈ (T_0-R^sp,T_0]Tail_p-1,s p(u(·,t);x_0,R)^p-1 + R^spf_L^∞(Q_R,R^sp(x_0,T_0)) +1.

By the triangle inequality, this implies the desired inequality (<ref>).




§ TIME REGULARITY

 In this section, we prove Hölder continuity in the time variable, assuming Hölder regularity in the spatial variable. For u∈ L^1(Q_R,r(x_0,t_0)), we will use the notation

    u_(x_0,t_0),R,r = _Q_R,r(x_0,t_0) u  x  t.

When the center (x_0,t_0) is clear from the context, we simply write u_R,r. 
   
    
        Let 1<p<2, 0<s<1 and Θ(s,p)=min(sp/p-1,1). Suppose that u is a local weak solution of 

    ∂_t u+(-Δ_p)^s u = f in  B_2× (-2^sp,0]

as in Definition <ref>,  where 

    u_L^∞(B_1 ×(-1,0])≤ 1,   sup_t ∈(-1,0]Tail_p-1,s p(u(·,t);0,1) ≤ 1,    and  f_L^∞(B_1 × (-1,0])≤ 1.

 Further, assume that for some δ∈(s,Θ(s,p)), there exists a positive constant K_δ depending on δ such that

    sup_t∈(-2^-sp,0][u(·,t)]_C^δ(B_1/2)≤ K_δ.

Then there is a constant C= C(N,s,p,K_δ,δ)>0 such that 

    u(x,t)-u(x,τ)≤ C t-τ^γ,     for all  x∈ B_1/4 and for all  t,τ∈ (-4^-sp,0]

where 

    γ = 1/s pδ+(2-p).

In particular, u∈ C^γ_t(Q_1/4,4^-sp).
    
    
   Take (x_0,t_0)∈ Q_1/4,4^-sp and choose 

    0<r<1/8,     0<θ<1/2(2^-sp - 4^-sp).

By construction,

    Q_r,θ(x_0,t_0)⊂ B_3/8×(-2^-sp,0].

Let η∈ C_c^∞(B_r/2(x_0)) be a non-negative smooth function, such that

    η≡η_L^∞(B_r/2(x_0)) B_r/4(x_0),    η_r=1/|B_r(x_0)|∫_B_r(x_0)η  x=1          ∇η_L^∞(B_r/2(x_0))≤C/r,
 
for some constant C=C(η_L^∞(B_r/2(x_0)),N)>0. Note that since the average of η is 1, we have the estimate

    η_L^∞(B_r/2(x_0))=1/|B_r/4(x_0)| ∫_B_r/4(x_0)η  x≤|B_r(x_0)|/|B_r/4(x_0)| η_r=4^N.

By the triangle inequality,

    _Q_r,θ(x_0,t_0)|u(x,t)-u_r,θ|  x  t   ≤_Q_r,θ(x_0,t_0)|u(x,t)-(u η)_r(t)|  x  t
       +_Q_r,θ(x_0,t_0)|u_r,θ-(u η)_r,θ|  x  t 
       + _Q_r,θ(x_0,t_0)|(u η)_r,θ-(u η)_r(t)|  x  t
        =: A_1+A_2+A_3,

where

    (u η)_r(t) = _B_r(x_0)u(y,t) η(y)  y.

Following the same steps in the proof of <cit.> we arrive at

    A_2 ≤ A_1+A_3,


    A_1≤ C K_δ  r^δ,      C=C(N,s,p)>0,

and 

    A_3≤sup_T_0,T_1∈ (t_0-θ,t_0]|(u η)_r(T_0) - (u η)_r(T_1)|.

For T_0,T_1 ∈ (t_0-θ,t_0] with T_0<T_1, we use the weak formulation (<ref>) with ϕ(x,t)=η(x), to obtain

    |B_r(x_0)| |(u η)_r(T_0)    - (u η)_r(T_1)| = |∫_B_r(x_0)u(x,T_0) η(x)  x - ∫_B_r(x_0)u(x,T_1) η(x)  x|
       =|∫_T_0^T_1∬_ℝ^N×ℝ^NJ_p(u(x,τ)-u(y,τ)) (η(x)-η(y))μ(x,y)τ|
       ≤|∫_T_0^T_1∬_B_r(x_0)× B_r(x_0)J_p(u(x,τ)-u(y,τ)) (η(x)-η(y)) μ(x,y) τ|
       + 2 |∫_T_0^T_1∬_(ℝ^N∖ B_r(x_0))× B_r/2(x_0)J_p(u(x,τ)-u(y,τ)) η(x)μ(x,y)τ|
        + | ∫_T_0^T_1∫_B_r(x_0) f(x,τ)η(x)  x τ| 
        = J_1 + J_2 + J_3.

For J_1, the estimate performed in the proof of <cit.> still applies and gives

    J_1≤ C K_δ^p-1θ r^N+ δ(p-1)-sp,

for some constant C=C(N,s,p,δ)>0. Regarding J_2, we may follow the steps in the proof of  <cit.> and obtain

    J_2 ≤ C θ r^N ( 1 + r^δ(p-1) -sp) ≤ Cθ r^N+δ(p-1)-sp,

for some constant C=C(N,s,p,δ,K_δ)>0.
Finally, we estimate J_3 as follows

    J_3    ≤∫_T_0^T_1∫_B_r(x_0)f(x,τ)η(x)  x  t ≤f_L^∞(B_r(x_0)×(T_0,T_1))∫_T_0^T_1∫_B_r(x_0)η(x)  x  t
       ≤ (T_1-T_0) B_r(x_0) f_L^∞(B_1×(-1,0))_B_r(x_0)η(x)  x ≤ C(N) θ r^N .

The combination of (<ref>)–(<ref>) implies

    A_3 ≤ C(1+K_δ^p-1) θ(1+ r^δ(p-1) -sp) ≤ C(1+K_δ^p-1) θ r^δ(p-1) -sp.

Therefore, by (<ref>) together with (<ref>)

    _Q_r,θ(x_0,t_0)|u(x,t)-u_r,θ|  x  t ≤ C( r^δ + θ r^δ(p-1) -sp),

for some constant C=C(N,s,p,δ,K_δ)>0.

We now split the rest of the proof into two cases.

Case 1: sp +δ(2-p) ≥ 1. Choose

    θ = 1/2(1/2^sp - 1/4^sp)r^sp+δ(2-p) .

Then (<ref>) reads

    _Q_r,θ(x_0,t_0)u(x,t)-u_r,θ x  t ≤ Cr^δ,

for some constant C=C(N,s,p,δ,K_δ)>0. 

Now we use the characterization of the Campanato spaces in ^n+1 with a general metric in <cit.>, see also <cit.>. Our setting does not fit directly in the context considered there, since we only work with cylinders that are one sided in the time direction, that is (t-r^sp+δ(2-p),t]× B_r(x) instead of (t-r^sp+δ(2-p),t+r^sp+δ(2-p))× B_r(x). Still, if one follows the proof in <cit.> with small modifications, one can also conclude the result in this setting. By <cit.>, u is δ-Hölder continuous in Q_1/4,1/4^sp with respect to the metric

    d((x,τ_1),(y,τ_2))= max{x-y , τ_2 - τ_1^1/sp+δ(2-p)}.

Note that since sp+ δ(2-p)≥ 1, d is a metric. The balls of radius r for this metric are of the form (t-r^sp+δ(2-p),t+r^sp+δ(2-p))× B_r(x).

In particular, for any τ_1 , τ_2 ∈ (-1/4^sp , 0] we have the estimate

    sup_x ∈B_1/4u(x,τ_1)-u(x,τ_2)≤ C τ_1-τ_2^γ,

where the constant C= C(δ, K_δ ,N, s, p)>0, and

    γ = δ/sp + δ(2-p) = 1/sp/δ + 2-p.

Case 2: sp +δ(2-p)<1. In this case, we make the choice

    r = 1/2(1/2^sp - 1/4^sp) θ^1/sp + δ(2-p).

From (<ref>) we obtain

    _Q_r,θ(x_0,t_0)u(x,t)-u_r,θ x  t ≤ C θ^δ/sp+δ(2-p).

Again, by <cit.> we obtain that u is δ/sp+δ(2-p)-Hölder continuous, in Q_1/4,1/4^sp with respect to the metric 

    d̃=max{x-y^sp+δ(2-p), τ_1-τ_2}.

As sp+δ(2-p)<1, d̃ is indeed a metric. In particular, we have the estimate

    sup_x ∈B_1/4u(x,τ_1)-u(x,τ_2)≤ C τ_1-τ_2^γ,

for some constant C= C(δ, K_δ ,N, s, p)>0, where 

    γ = δ/sp+δ(2-p).

    



§ PROOF OF THE MAIN THEOREM

We are now ready to give the proof of our main theorem, namely Theorem <ref>, that encompasses both the spatial regularity and regularity in time.

 The proof follows from a combination of Theorem <ref>, Remark <ref> and Proposition <ref>. We spell out the details below.
It remains to prove the regularity in time. We  assume x_0=0 and T_0=0 and argue as in the proof of Theorem <ref>. Let 

    u_R(x,t):=1/M u(RM^p-2/sp x+y_0,R^spt),    (x,t)∈ B_2× (-2^sp,0]

where y_0 is chosen so that 

    1/2 RM^p-2/sp+|y_0|≤  R/2

and

    M=   (2+(2 N ω(N)/sp)^1/p-1)u_L^∞(B_R× (-R^sp,0]) +2^1+N+spsup_t∈ (-R^sp,0]Tail_p-1,s p(u(·,t);0,R)^p-1
       + R^spf_L^∞(B_R×(-R^sp,0]) +1.

Estimate (<ref>) implies

    2RM^p-2/sp+|y_0|≤ 2R.

Therefore, u_R is a local weak solution of ∂_t u+(-Δ_p)^s u= f̃ in B_2× (-2^sp,0], where

    f̃(x,t):= R^sp/M f(RM^p-2/spx +y_0 , R^spt).

With the notation R̃=RM^p-2/sp, it is straight forward to verify that (<ref>) implies

    u_R_L^∞(B_1× (-1,0])=M^-1u_L^∞(B_R̃(y_0)× (-R^sp,0])≤ M^-1u_L^∞(B_R× (-R^sp,0])≤ 1.

As in the proof of Theorem <ref> we also have

    sup_t∈ (-1,0]∫_ℝ^N∖ B_1|u_R(z,t)|^p-1/|z|^N+s p   dz≤ 1.

We also have

    f̃_L^∞(B_1×(-1,0])   =R^sp/M  f(RM^p-2/spx+ y_0, R^spt)  _L^∞(B_1× (-1,0])
       = R^sp/Mf_L^∞(B_R̃(y_0)×(-R^sp,0])≤R^sp/Mf_L^∞(B_R× (-R^sp,0])≤ 1.

The above bounds combined with Theorem <ref> and Remark <ref> imply

    sup_t∈(-2^-sp,0] [u_R(·,t)]_C^θ(B_1/2)≤ C,

for every θ∈(0,Θ) with C=C(N,s,p,θ)>0. By using this together with Proposition <ref>, we obtain

    sup_x∈ B_1/4[u_R(x,·)]_C^γ((-4^-sp,0])≤ C

where 

    γ = 1/s pθ+(2-p),
 and C=C(N,s,p,θ). Scaling back, (<ref>) and (<ref>) imply, (again with the notation R̃=RM^p-2/sp)

    sup_t∈ (-(R/2)^sp,0][u(x,t)]_C^θ(B_R̃/2(y_0))≤ CM^1+(2-p)/spR^-θ

and

    sup_x∈ B_R̃/4(y_0)  [u(x,·)]_C^γ((-4^-spR^sp,0])≤ CMR^-γ

for all y_0 that satisfies (<ref>), with the constant C=C(N,s,p,θ)>0. Therefore, we obtain the estimates (<ref>) and (<ref>) in the whole B_R/4 by varying y_0 as in the proof of Theorem <ref>. Since this holds for all θ<Θ and with γ = (s p/θ+(2-p))^-1 which converges to Γ as θ→Θ, this implies the desired result.




§ ACKNOWLEDGEMENT
 The first author is supported by the seed grant from IISER Berhampur. E.L and A.T have been supported by the Swedish Research Council, grants
no. 2023-03471 and 2016-03639.
plain



0.1cm
Prashanta Garain

Department of Mathematical Sciences

Indian Institute of Science Education and Research Berhampur

Berhampur, Odisha 760010, India

E-mail address: pgarain92@gmail.com 



Erik Lindgren

Department of Mathematics

KTH - Royal Institute of Technology
100 44, Stockholm, Sweden

E-mail address: eriklin@math.kth.se 



Alireza Tavakoli

Department of Mathematics

KTH - Royal Institute of Technology
100 44, Stockholm, Sweden

E-mail address: alirezat@kth.se 




