A PERTURBATIVE APPROACH TO H ¨ OLDER CONTINUITY OF SOLUTIONS TO A NONLOCAL p -PARABOLIC EQUATION

. We study local boundedness and H¨older continuity of a parabolic equation involving the fractional p -Laplacian of order s , with 0 < s < 1, 2 ≤ p < ∞ , with a general right hand side. We focus on obtaining precise H¨older continuity estimates. The proof is based on a perturbative argument using the already known H¨older continuity estimate for solutions to the equation with zero right hand side.


Introduction
In this paper, we study the local boundedness and Hölder regularity of solutions to the inhomogeneous equation where f ∈ L r loc (I; L q loc (Ω)) with q ≥ 1, r ≥ 1, p ≥ 2 and s ∈ (0, 1).Here, (−∆ p ) s is the fractional p-Laplacian, arising as the first variation of the Sobolev-Slobodeckiȋ seminorm (−∆ p ) s u(x) := 2 P.V.
Nonlocal equations involving operators of the type above, with a singular kernel, were first considered in [IN10] to the best of our knowledge.In this study, continuing the work in [BLS21], we perform a perturbative argument to obtain Hölder continuity estimates, with explicit exponents for the equations with a right hand side.Our approach closely follows the arguments in [TU14] and [BLS18].In such perturbative arguments it is often possible to establish Hölder regularity results for bounded solutions using only L ∞ estimates for the equations with zero right hand side.This is not the case here.Due to the presence of a suprimim in time in the tail (see section 3), we are led to proving a L ∞ bound for equations with right hand sides, this is Theorem 1.1.The proof is inspired by the work [AS67].
Below, we state the main results.For the definition of the tail and relevant function spaces, see Section 2. We use the following notation of parabolic cylinders The exponent p ⋆ s = np n−sp is the critical exponent for the Sobolev embedding therem, see Proposition 2.5.We denote by p ′ , the Hölder conjugate of p, that is p ′ = p p−1 .
The local boundedness of the solutions to equations modeled on (1.1) with zero right hand side was obtained in [S19].The results concern operators of the form where K is symmetric in the space variables and satisfies the ellipticity condition Later in [DZZ21] local boundedness for certain right hand sides of the form f (x, t, u) was established.[S19(2)] contains a Harnack inequality for equations with zero right hand side.Hölder regularity has also been established in [APT22] and [L22] for all 1 < p < ∞ for equations with zero right hand sides.In [BLS21] they prove Hölder continuity of the solutions with explicit exponents (for f = 0 and K = |x − y| −n−sp ).Recently in [GDS22], the same type of result has been established for nonlocal equations with double phase, that is for diffusion operators involving two different degrees of homogeneity and differentiability.
In this study, continuing the work in [BLS21], we perform a perturbative argument to obtain Hölder continuity estimates with explicit exponents for equations with a right hand side.
1.1.1.Comparison of the results to some previous works.Local boundedness and continuity.We compare our boundedness result to [DZZ21].Their result concerns more general right hand sides depending on the solution as well.In the limiting case of s → 1, they reproduce the local boundedness result contained in [DiB93] for the evolution p-Laplacian equation.To compare the results, if we restrict their result to right hand sides that are u-independent, their assumption on the integrability becomes q, r > n+sp sp ( p(n+2s) 2sp+(p−1)n ).Their analysis is done with the same integrability assumption in time and space; Our local boundedness result, Theorem 1.1, contains this range of exponents.
In the limiting case when s goes to 1, our assumptions become q ≥ (p ⋆ ) ′ , r ≥ p ′ , and 1 − 1 r − n pq > 0. This is in accordance with the classical condition for boundedness of the evolution p-Laplace equation, see for example Remark 1 in [LSS13].If we assume the same integrability in time and space, the condition 1 − 1 r − n pq > 0 reduces to f ∈ L q with q > n+p p .This matches the condition in [Ve93].Now we turn our attention to the nonlocal elliptic (time independent) case.For r = ∞ the condition for boundedness and basic Hölder continuity becomes q > n sp , if sp < n and q > 1 , if sp ≥ n .
In the case sp < n, this is the same condition for local boundedness and continuity contained in [BP16].When sp > n and q ≥ 1, the boundedness and Hölder continuity for the time independent equation is automatic using Morrey's inequality.Our result does not cover the case of r = ∞ , q = 1 , which one would expect in comparison to the time independent case.Hölder continuity exponent : In the case r = ∞, the critical Hölder continuity exponent min Θ, r(spq − n) − spq q(r(p − 1) reduces to min {Θ, sp p−1 (1 − n spq )} which matches the results in [BLS18].Let us also compare our results to the local p-parabolic equation studied in [TU14] where precise Hölder continuity exponents are obtained.If we send s to 1, (1.4) becomes min {1, r(pq − n) − pq q(r(p − 1) which is in accordance with the result in [TU14].
In [GDS22] explicit Hölder continuity exponents for the more general case of double phase nonlocal diffusion operators were obtained.The ideas explored there are similar to the ones in [BLS21], but their result allows for a bounded right hand side instead of just zero.Their result implies the Hölder continuity exponent that we get in the case of f ∈ L ∞ , although with a slightly different estimate of the Hölder constants.
1.2.Plan of the paper.In Section 2 we introduce some notations and preliminary lemmas.We also restate and adapt a result on the existence of solutions to our setting.
In Section 3, we establish basic local Hölder regularity and boundedness for local weak solutions.Section 4 is devoted to proving Theorem 1.2.A so called tangential analysis is performed to get specific Hölder continuity exponents in terms of q, r, s, and p.
The article is also accompanied by two appendices.In the first one, Appendix A, we work out the details for a modified version of [BLS21, Theorem 1.1].The aim is to establish a Hölder estimate in terms of the tail quantity.
In Appendix B we justify using certain test functions in the weak formulation of (1.1).
We use the notation B R (x 0 ) for the open ball of radius R centered at x 0 .If the center is the origin, we simply write B R .We use the notation of ω n for the surface area of the unit n-dimensional ball.For parabolic cylinders, we use the notation Q r,r θ (x 0 , t 0 ) := B r (x 0 ) × (t 0 − r θ , t 0 ].If the center is the origin, we write Q r,r θ .
We will work with the fractional Sobolev space extensively: We also need the space W β,q (Ω) for a subset Ω ⊂ R n , defined by In the following, we assume that Ω ⊂ R n is a bounded open set in R n .We define the space of Sobolev functions taking boundary values g where Ω ′ is an open set such that Ω ⋐ Ω ′ .We recall the definition of tail space which is endowed with the norm , plays an important role in regularity estimates for solutions to fractional problems.
Let I ⊂ R be an interval and let V be a separable, reflexive, Banach space endowed with a norm q V .We denote by V ⋆ its topological dual space.Let us suppose that v is a mapping such that for almost every t ∈ I, v(t) belongs to V .If the function t → v(t) V is measurable on I and 1 ≤ p ≤ ∞, then v is an element of the Banach space L p (I; V ) if and only if By [Sh97, Theorem 1.5], the dual space of L p (I; V ) can be characterized according to (L p (I; V )) ⋆ = L p ′ (I; V ⋆ ).We write v ∈ C(I; V ) if the mapping t → v(t) is continuous with respect to the norm on V .2.2.Pointwise inequalities.We will need the following pointwise inequality: Let p ≥ 2, then for every A, B ∈ R we have (2.1) For a proof look at [BLS21, Remark A.4], a close inspection of the proof reveals that the constant can be taken as C = 3 • 2 p−1 .Before stating the next inequality, we recall [BP16, Lemma A.2].
Proof.First notice that using (2.1) for a − b − c + d = 0: after verifying the trivial case a − b − c + d = 0, we get the inequality (2.3) Using (2.3) in the inequality above concludes the proof.
2.3.Functional inequalities.We need the following basic inequalities for the tail.
where ω n is the measure of the n-dimensional open ball of radius 1.
We also recall the following Sobolev and Morrey type inequalities: Proposition 2.5.Suppose 1 < p < ∞ and 0 < s < 1.Let Ω ⊂ R n be an open and bounded set.Define p ⋆ s as (2.4) In particular the following Poincaré inequality holds true for some C = C(n, s, p).
Remark 2.6.The Sobolev type inequalities above are also valid for functions u ∈ X s,p 0 (Ω, Ω ′ ), where Ω is a bounded open set and Ω ′ is an open set such that Ω ⋐ Ω ′ .This can be seen using the fact that there is an extension domain containing Ω and included in Ω ′ .
Lemma 2.7.If w is contained in L q1,r1 (Ω × J) ∩ L q2,r2 (Ω × J), then w is contained in L q,r (Ω × J), where Moreover, The following three lemmas will be needed in the proof of our local boundedness result, Proposition 3.3.
Lemma 2.8.Let sp < n and assume that w is in Moreover, In particular, in the case of 1 r + n spq = 1 we have Proof.Consider a pair of exponents r = ( 1 r ′ − (1 − 1 r − n spq )) −1 = spq n , and q = q ′ such that 1 r ′ + n spq ′ = 1.Using Hölder's inequality (2.9), we obtain Now we use Lemma 2.7 with the choice This yields ) .The relations above hold for λ = 1 r = n spq ′ and using Young's inequality we get . This concludes the desired result.
Lemma 2.9.Let sp > n and assume that w Then w belongs to L pq ′ ,pr ′ (Q R,R sp ) and Proof.We use Lemma 2.7, with the choice which holds for λ = 1 r ′ = 1 − 1 q ′ .This yields Therefore, using λ = 1 r ′ = 1 q we arrive at .
Using Young's inequality, we conclude Lemma 2.10.Let sp = n, q ≥ 1, and r ≥ 1 such that Then w belongs to L pq ′ ,pr ′ (Q R,R sp ) and Proof.We use Lemma 2.7 with the choice 1 pr ′ = λ p and 1 Due to the assumption 1 l = r ′ p (1 − 1 r − 1 q ), the above equalities hold for λ = 1 r ′ .Hence we get Using Young's inequality for the right hand side, we can conclude 2.4.Weak solutions.
Definition 2.11.For any t 0 , t 1 ∈ R with t 0 < t 1 , we define We say that u is a local weak solution to the equation and it satisfies (2.10) for any ϕ ∈ L p (J; W s,p (Ω)) ∩ C 1 (J; L 2 (Ω)) which has spatial support compactly contained in Ω.In equation (2.10), the symbol q , q stands for the duality pairing between W s,p (Ω) and its dual space (W s,p (Ω)) * .Now, we define the notion of a weak solution to an initial boundary value problem.
Suppose also that Then for any initial datum g 0 ∈ L 2 (Ω), there exists a unique weak solution u to problem (2.12) Proof.In [BLS21, Theorem A.3] the same result is proven with a stronger condition g t ∈ L p ′ (I; W s,p (Ω ′ ) ⋆ ).The stronger condition, is not needed in the proof.This condition can be replaced with g t ∈ L p ′ (I, X s,p 0 (Ω; Ω ′ ) ⋆ ) in all of the steps in the proof, except that the construction gives us a C(I; L 2 (Ω)) solution.There, the stronger assumption is used only to show that the boundary condition is in C(I; L 2 (Ω)), which we assume here.

Basic Hölder regularity and stability
Throughout the rest of the article, we assume 0 < s < 1 and 2 ≤ p < ∞.
Here, we argue that the norm of the (s, p)-caloric replacement of u is close to u if f is small enough.By the (s, p)-caloric replacement of u in a cylinder B ρ (x 0 ) × I we mean the solution to the following Here τ 0 is the initial point of the interval I. First we show the existence of a (s, p)-caloric replacement using Theorem 2.13 Proposition 3.1.Let u be a local weak solution of u t + (−∆ p ) s u = f in the cylinder B σ × J, for some interval J = (t 1 , t 2 ] with f ∈ L q,r loc (B σ × J), for q > (p ⋆ s ) ′ and r > p ′ .In addition, we assume that u ∈ L p (J; L p−1 sp (R n )).Then for any 0 < ρ < σ, and closed interval I ⋐ J, the (s, p)-caloric replacement of u in B ρ (x 0 ) × I (weak solution to (3.1)) exists.
Proof.We shall check the conditions in Theorem 2.13.If they are satisfied there exists a unique weak solution v ∈ L p (I, W s,p (B σ )) ∩ L p−1 (I; L p−1 sp (R n )) ∩ C(I; L 2 (B ρ )) to the problem (3.1).The only condition on u that is not immediate from the fact that u is weak solution is ∂ t u ∈ L p ′ (I; X s,p 0 (B ρ , B σ ) ⋆ ).We have to show that for every function ψ ∈ L p (I; X s,p 0 (B ρ , B σ )) We shall verify this for test functions belonging to the dense subspace, ψ ∈ L p (I; X s,p 0 (B ρ , B σ )) ∩ C 1 0 (I; L 2 (B)).We use the equation to do so.We have By Hölder's inequality, we have For the other nonlocal term, we note that for every x ∈ B ρ and Therefore, For the other term, )), we get by Hölder's inequality (3.5) Therefore, combining with (3.2) , (3.3), and (3.4) we obtain ) , with C = C(n, s, p, q) also depending on q.
Proof.Let J := [T 0 − ρ sp , T 0 ], throughout the proof, we drop the dependence of the balls on the center and write B ρ instead of B ρ (x 0 ), and By subtracting the weak formulation of the equations (2.10) for u and v with the same test function Now we take ϕ := u − v, which belongs to L p (J; X s,p 0 (B ρ ; B σ )) but it may not be in C 1 (J; L 2 (B ρ )).We justify taking this as a test function in Appendix B. By Proposition 6.1 with (3.8) where in the third line we have used u(x, T 0 − ρ sp ) = v(x, T 0 − ρ sp ).The left hand side is essentially the W s, p seminorm.By the pointwise inequality (2.1) Therefore, by (3.8) and Hölder's inequality (3.9) Now we consider three cases: sp < n, sp > n,and sp = n.
Case sp < n.By Hölder's inequality (2.9) and Sobolev's inequality (2.5) we have (3.10) Combined with (3.9) this yields where C = C(n, s, p).By the Poincaré inequality Also from (3.11) and (3.10) we get Case sp > n.In this case, we use Morrey's inequality (2.6) and Hölder's inequality and obtain (3.12) Together with (3.9), this implies By the Poincaré inequality Combining (3.12) and (3.13), we get Case sp = n.In this case, we use the critical case of Sobolev's inequality (2.7) for l = q ′ and obtain Hence using Hölder's inequality, we have for any r The constant C = C(n, s, p, q) above does blow up as q goes to 1.In a similar way as in the prior cases, we get for q > 1 and Here in the case of sp = n, and in the case sp = n, Next, we perform a Moser iteration to get an L ∞ bound for the difference between the solution and its (s, p)-caloric replacement.
Proposition 3.3.Let u be a local weak solution of Then in the case of sp = n, we have , In the case of sp = n, given any l such that p r ′ (1 , Proof.Throughout the proof we write Q R,R sp instead of Q R,R s,p (x 0 , T 0 ).We test the equations with powers of u − v and perform a Moser iteration.Using Proposition 6.1 with and (3.15) In the last line, we have used Hölder's inequality.Here In particular for 0 ≤ t ≤ M Hence Using Lemma 2.2 for the second term in the left hand side of (3.15), and (3.16) in the first term we obtain We consider three cases depending on whether sp > n, sp = n, or sp > n.
Case sp < n: Using Sobolev's inequality in the second term in (3.17) and applying (3.18) we get Upon multiplying both sides by 3 (3.20) By replacing the constant C(n, s, p) above in Sobolev's inequality with max {1, C(n, s, p)}, we can assume C(n, s, p) ≥ 1.Using this and that δ ≥ 1, and (β+p−1) p β(β+1) ≥ 1 we obtain Using this together with (3.20), and (3.18) we get where C = C(n, s, p).Recalling our choice of δ, (3.14), in the case of δ > 1 and in the case of δ = 1 Now notice that since ν > 0, if we take ϑ = 1 + spν n , the exponents (ϑr ′ ) ′ , (ϑq ′ ) ′ satisfy the condition of Lemma 2.8.Indeed, Using Lemma 2.8 for the exponents (ϑq ′ ) ′ and (ϑr ′ ) ′ we get Now we iterate this inequality with the following choice of exponents With the notation Iterating this yields Since ϑ > 1, we have the following convergent series and (3.26) Inserting (3.26) to (3.25) and sending n to infinity we obtain Case sp > n.Here we use Morrey's inequality (2.6) for the second term in (3.17).Instead of (3.19) we obtain δ 2(β + 1) Following the same steps as in the case sp < n we arrive at and we can apply Lemma 2.9 with the exponents (ϑq ′ ) ′ and (ϑr ′ ) ′ .This gives (3.27) We apply (3.27) with the exponents Let By iterating the above inequality, we get (3.28) Since ϑ > 1, we have the following convergent series Inserting this into (3.28), and sending n to infinity we get . Hence, we arrive at the desired estimate Case sp=n.Here we use the critical case of Sobolev-Morrey inequality, (2.7) with This applied for the second term in (3.17) implies Following the same step as in the previous two cases, we arrive at Notice that due to the choice of l, (3.29), we have ϑ > 1.Then the exponents (ϑr ′ ) ′ and (ϑq ′ ) ′ satisfy Therefore, we can apply Lemma 2.10 with the exponents (ϑr ′ ) ′ and (ϑq ′ ) ′ to get (3.30)We apply (3.30) with the exponents Let By iterating the above inequality, we get Since ϑ > 1, we have the following convergent series Inserting this into (3.30), and sending n to infinity we get Hence we arrive at the desired estimate Notice that −u is a solution to the same type of problem, and we can apply the above proposition to −u.Since −v is the (s, p)-caloric replacement of −u we get the same bound on ′ in the case sp < n, and 1 r + 1 q < 1 and q > 1 in the case sp ≥ n.
If sp = n, then for any l such that p r ′ (1 Now we combine the local boundedness results for the equations with zero right hand side (see [S19] and also [DZZ21]) with Proposition 3.3 to prove local boundedness for the equation with nonzero right hand side.
Proof of Theorem 1.1.For u, a local weak solution of Tail p−1,sp v( q , t); x 0 , R 2 with C depending on n, s and p.Therefore, ) sp (x0,T0)) |u| p dx dt Using Lemma 2.3 in (3.32) we arrive at +2 sup Tail p−1,sp u( q , t); x 0 , R 2 , (3.33) where C = C(n, s, p).Finally, using Proposition 3.3 to estimate the term u − v L ∞ (Q R,R sp ) , in (3.33) we get the desired result.Here the estimate is written in the case sp ′ in the case sp < n, and 1 r + 1 q < 1 and q > 1 in the case sp ≥ n.
Using Corollary 3.4 we get: Tail p−1,sp (u( q , t); x 0 , R) p + C(n, s, p) ϑ , with ϑ and ν defined in Corollary 3.4, here the estimate is only written in the case sp = n for simplicity.Since Q R,R sp (x 0 , T 0 ) ⊂ Q R1,R sp 1 (z, T 1 ) the above expression is less than Concerning the tail term , since B R (x 0 ) ⊂ B R1 (z), using Lemma 2.4 we have Tail p−1,sp (u( q , t); x 0 , R) and by the choice of the radii, we have Hence, taking the supremum in time and using Minkowski's inequality in (3.37), we arrive at sup Tail p−1,sp (u( q , t); z, R 1 ) p , where the constant C above depends on n, s, and p.In conclusion, Tail p−1,sp (u( q , t); z, R 1 ) p + ϑ Now we make the choice ρ = R θ 2 with θ := 1 + ξ δp + n + sp .

2
, where We can then conclude that for any cylinder of arbitrary size we have Now we use the characterization of the Campanato spaces in R n+1 with a general metric in [G09], see also [DaP65].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 , t] × B r (x) instead of (t − r sp , t + r sp ) × B r (x).Still, if you follow the proof in [G09] with small modifications, you can also conclude the result in this setting.
In the case of sp ≥ 1, using [G09, Theorem 3.2] we get the the Hölder continuity of u with exponent ζ in Q σR,(σR) sp with respect to the metric for which the balls of radius r are of the form (t − r sp , t + r sp ) × B r (x).which means In the case of sp < 1 we use the metric The balls of radius r are of the form (t − r, t + r) × B < 1 and q ≥ (p ⋆ s ) ′ in the case sp < n, and 1 r + 1 q < 1 and q > 1 in the case sp ≥ n.
Let u be a local weak solution to the equation Tail p−1,sp (u( q , t); 0, R) ≤ M, and and δ M,R,σ (ω) converges to 0 as ω goes to 0.
Proof.The existence of such a bound follows immediately from Corollary 3.4.
To show the convergence of τ M,R,σ to zero we argue by contradiction, suppose that there is a sequence f n ∈ L q,r (Q R,R s p ) and u n such that Tail p−1,sp (u n ( q , t); 0, R) ≤ M and f n L q,r (Q R,R s p ) → 0, but Using (3.11) from Lemma 3.2, we have By assumption, (3.40)By Corollary 3.4 ) is uniformy bounded, (3.41) and (3.40) gives us a uniform bound on ϕ n L ∞ (Q R,R s p (x0,T0)) .Now we are in a position to use Theorem 3.5 for both of the sequences u n and ϕ n , which gives us a uniform bound on the Hölder seminorms of u n and ϕ n in Q σR,(σR) sp .Therefore, by Arzela-Ascoli's Theorem u n − ϕ n has a uniformly convergent subsequence in Q σR,(σR) sp .By (3.39) the limit is 0, contradicting (3.38).
Then there exists ω such that if 2 sp with exponents α space and α sp−(p−2)α in time, as long as . (4.1) Step 1: Decay at the origin.
For this part, we prove a decay at the origin for u under the assumptions Tail p−1,sp (u; 0, 1) ≤ 1, and f L q,r (Q1,1) ≤ ω. (4.2) We introduce the parabolic cylinder with β = sp − (p − 2)α.We show that for any exponent α satisfying (4.1), the following holds for r < 1 It is enough to prove the inequality for a sequence of r = λ k , (k) ∞ 0 , for some λ < 1.Without loss of generality, we assume u(0, 0) = 0. Consider the rescaled functions with λ small enough to be determined later.We will prove the following by induction, For k = 0, (4.3) follows from our assumptions (4.2).
Therefore, using (4.3), for v k we have By Corollary 3.4 This is a bound independent of k.We can take ω to be less than 1 and take C 1 = C(n, s, p)(3 + 2C(n, s, p, q, r), with the C(n, s, p, q, r) coming from Corollary 3.4, so that the constants C 1 , C 2 are independent of ω as well.Now we proceed and prove (4.3) for k + 1.First, we state our choice of . (4.5) Since λ < 1 4 , and Notice that βΓ ≥ Θ, by the above choice of β.Thus, and by the assumption (4.5) This is possible since δ(ω) converges to zero as ω → 0.Then, (4.6) implies which is the first part of (4.3).For the second part, we want to show sup We split the integral into three parts.Using the induction hypothesis sup For remaining part, we transfer the estimate (4.4) to v k+1 and obtain Using the assumption (4.5) on λ, we obtain sup Step 2: Regularity in a cylinder.We choose α as in (4.1) and let ω be as in Step 1.For a point (x By the choice of L, ũ satisfies the conditions (4.2) in Step 1.Since L ≥ 1 we immediately have As for the L q,r norm of f we have Here we have used 1 − 1 r − n spq > 0. Notice that in the case of sp ≥ n, we are assuming 1 − 1 r − 1 q > 0 which is a stronger assumption.Now we verify the assumption on the tail.sup Tail p−1,sp (u( q , t); 0, 1)

Now we can apply
Step 1 to ũ and we get the decay In terms of u, this means Now take two points (x 1 , t 1 ) , (x 2 , t 2 ) ∈ Q 1 2 , 1 2 sp and split the line joining them into 1 + [L p−2 ] pieces, say (y i , τ i ) Now we prove the Hölder regularity at any scale.
Proof of Theorem 1.2.We will consider the rescaled functions Tail p−1,sp (u( q , t); x 0 , R) , where ω = ω(n, s, p, q, r, α) is the same as in the proof of Proposition 4.1 and ι 2 ) sp (x 0 , T 0 ) by varying ι over.Note that for these choices of We now verify that ũι satisfies the conditions of Proposition 4.1.The L q,r norm of the right hand side is Tail p−1,sp (u( q , t); x 0 , R) ≤ 1.

Appendix A
In this section, we spell out the necessary modifications to prove the following theorem 5.1 which is a modified version of [BLS21, Theorem 1.2].
Proof.In the proof of [BLS21, Proposition 4.1], the L ∞ (R n × [0, 1]) boundedness is only used in Step 3, in the estimation of the nonlocal terms I 2 and I 3 , which are defined by We also recall the definition of Ĩ2 and Ĩ3 Ĩi := The general argument is the same but instead of using the L ∞ norm of u(y) we can keep the inequality as it is and write and for u h Here we have used B R (h) ⊂ B 1 , and . Using this we get and we can conclude Which is the same as equation (4.6) in [BLS21].
We can estimate the W s,p semi-norm of a solution as follows.The proof follows the argument in [BLS21, Lemma 7.1].
Proof.Without loss of generality, we may suppose that x 0 = 0. Let Then u is a local weak solution in B 2 × (−2 R s p , 0] and u ≥ 1 in B 2R × [−R s p , 0].We choose ϕ and ψ exactly as [BLS21, Lemma 7.1], that is and Then for ϕ(x, t) = η(x)ϕ(t) we get 0 The only difference in the proof is in estimating the term sup (u(y, t) + ) p−1 |x − y| n+s p dy.
Noticing that for We can now prove the following modified version of [BLS21, Theorem 4.2].
Theorem 5.4 (Spatial almost C s regularity).Let Ω ⊂ R n be a bounded and open set, I = (t 0 , t 1 ], p ≥ 2 and 0 < s < 1. Suppose u is a local weak solution of . Then u ∈ C δ x,loc (Ω × I) for every 0 < δ < s.More precisely, for every 0 < δ < s, R > 0 and every (x 0 , T 0 ) such that there exists a constant C = C(n, s, p, δ) > 0 such that Tail p−1,sp (u; x 0 , 2R) (5.5) Proof.The proof is essentially the same as the proof of [BLS21, Theorem 4.2].We assume for simplicity that x 0 = 0 and T 0 = 0, and set Notice that by Lemma 5.3 we have Then u R,α (x, t) is a local weak solution of Tail p−1,sp (u( q , t); 0, 1) ≤ 1, This function satisfies the assumption of Proposition 5.2, and we can do the same argument as in [BLS21] to obtain sup for a C independent of α and by scaling back we get , 0] we get the desired result.
We now address the improved regularity and start with the following modified version of [BLS21, Proposition 5.1].
Proof of Theorem 5.1.Consider a cylinder Q 2ρ,2ρ sp (x, τ ) ⋐ Ω × I, first, we prove the following type of bound on the Hölder seminorm in Q ρ/4,ρ s p /4 (x, τ ), and later with the aid of a covering argument we conclude the claim of the theorem.Claim: For any (x 1 , τ 1 ), (x 2 , τ 2 ) ∈ Q ρ/4,ρ s p /4 (x, τ ) we have (5.9) The regularity in space variable has been proven in Theorem 5.6.To prove the part on time regularity we set and consider the rescaled functions ũρ,ι (x, t) Moreover, ũρ,ι (x, t) satisfies the conditions of Proposition 5.7.Indeed by construction Tail p−1,sp (ũ ρ,ι ; 0, 2) ≤ 1 and the estimate (5.8) follows from (5.7) in Theorem 5.6.From Proposition 5.7 we obtain sup with C = C(n, s, p, γ) for every 0 < γ < Γ(s, p).By scaling back this translates to (5.10) (in particular, this will be the case under the stronger assumption sup t∈I Tail p,sp (u( q , t); x 0 , R) < ∞ that we use in this article.)For any time interval In addition, assume that F is a globally Lipschitz function with F (0) = 0, which is either bounded or F (a) = a.
Then we have: where F (a) := a 0 f (t) dt is the primitive function of F .
Proof.The proof is essentially the same as [BLS21, Lemma 3], except that here we don't use a cut off function and don't have the global boundedness of u in the ball.For simplicity we assume x 0 = 0, R = 1 and σ = 2.For a function ϕ ∈ C((T 0 , T 1 ); L 2 (B)) ∩ L p ((T 0 , T 1 ); X s,p 0 (B, B 2 )), we use the following regularization of functions This regulization process gives us a test function ϕ ε ∈ C 1 ((T 0 +ε, T 1 −ε); L 2 (B))∩L p ((T 0 +ε, T 1 −ε); X s,p 0 (B, B 2 )).Let t 0 = T 0 + ε 0 and t 1 = T 1 − ε 0 and we test the equation with ϕ ε as above, for ε < ε0 2 .First, we will show the claim for the smaller interval [t 0 , t 1 ] ⊂ [T 0 , T 1 ], and then through a limiting argument, prove the result for the whole interval.As in equation (3.5) in [BLS21], we get and we obtain a similar identity for v without t1 t0 B ϕ ε f dx dt in the right hand side.Here Σ u is defined by Observe that by using an integration by parts, the term Σ u (ε) can be rewritten as where we also used that ζ has compact support in (−1/2, 1/2).By subtracting the identities for u and v, we obtain After an integration by parts we get We now wish to pass to the limit in I 1 , I 2 , and I 3 .Let w = u − v, we now treat I 1 .The fact that F is globally Lipschitz together with where C is the Lipschitz constant of F .After integrating and using Hölder's inequality, we obtain Using the traingle inequality we get using a computation similar to (6.3) we obtain and These two expressons converge to zero, since w ∈ C([T 0 , T 1 ], L 2 (B)) and (t 0 − ε, t 1 + ε) ⋐ (T 0 , T 1 ).This shows that I 1 converges to zero.By similar reasoning, I 2 also tends to zero.For the term I 3 , we have The sequence w ε is bounded in L p ⋆ s ,p (B × (t 0 , t 1 )), therefore, it has a weakly convergent subsequence.Using the pointwise convergence of w ε to w, we get the weak convergence of w ε − w to zero.By the assumptions on q, r together with Hölder's inequality (2.9), f (x, t) belongs to the dual space L (p ⋆ s ) ′ ,p ′ (B × (t 0 , t 1 )).Therefore, for a function g ∈ L p (t 0 − ε 0 , t 1 + ε 0 ) is continuous for −ε 0 ≤ a ≤ ε 0 .Hence, we get lim ε→0 w ε (x, t) L p ⋆ s ,p (B×(t0,t1)) = lim ε→0 w ε (x, t + εσ) L p ⋆ s ,p (B×(t0,t1)) = w L p ⋆ s ,p (B×(t0,t1)) .
Combined with the convergence of the norms, this implies the strong convergence in the norm, in particular, we have w ε (x, t + εσ) − w ε (x, t) L p ⋆ s (B) L p (t0,t1) → 0.
This shows that Θ 2 (ε) = t1 t0 B G(x, t) F (w ε (x, t)) ε − F (w(x, t) converges to zero.Finally, we let ε 0 go to zero to get the desired result for [T 0 , T 1 ].We need to show that the following converge to zero as ε 0 tends to 0. The arguments will be reminiscent of the ideas in the previous part.We start with J 2 , in the case of a bounded F , F is globally Lipschitz and we have w(x, T 1 ) − w(x, T 1 − ε 0 ) w(x, T 1 ) + w(x, T 1 − ε 0 ) dx ≤ w( q , T 1 ) − w( q , T 1 − ε 0 ) L 2 (B) w( q , T 1 ) + w( q , T 1 − ε 0 ) L 2 (B) .
Again since w ∈ C([T 0 , T 1 ]; L 2 (B)), this term converges to 0. J 1 can be treated in a similar way.For the term J 4 , using |F (a)| ≤ C|a| we get |w(x, t)||f (x, t)| dx dt.
Since w ∈ L p ⋆ s ,p (B × [T 0 , T 1 ]) and f ∈ L (p ⋆ s ) ′ ,p ′ (B × [T 0 , T 1 ]), using Hölder's inequality (2.9), one can see that w(x, t)f (x, t) ∈ L 1 (B × [T 0 , T 1 ]).Now using the absolute continuity of the integral for integrable functions we can conclude that J 4 converges to 0. The reasoning for convergence of J 3 is similar.Now we turn our attention to the nonlocal terms.This implies that the integrand involved in Θ 1 belongs to L 1 ([T 0 , T 1 ]; L 1 (B 2 ×B 2 )).And similar to the treatment of J 4 , since the volume of the integration region is shrinking to 0, Θ 1 converges to 0. To deal with Θ 2 , notice that F (w(x, t)) ∈ L p ([T 0 , T 1 ]; L p (B)) and define J p (u(x, t) − u(y, t)) − J p (v(x, t) − v(y, t)) |x − y| n+sp dy.
This concludes the result.N 1 can be treated in an exactly similar manner.

r 1 sp
(x).Hence we have a decay of order r ξ sp p of the average of u on the half balls.[G09, Theorem 3.2] implies the following Hölder continuity on Q σR1,(σR1) sp |u

)
for every 0 < µ < T 1 − T 0 .Here C depends on the n, h 0 , s, p, µ and β.Proof.The only major difference from the proof of Proposition 5.2 is in the estimation of term I 11 and it can be treated in the exact same way as in the proof of [BLS21, Proposition 5.1].Using the previous Proposition with the same type of modifications as in the proof of Theorem 5.4 we can state the following version of [BLS21, Theorem 5.2].
In particular, I wish to express my gratitude to the Department of Mathematics at Uppsala University for its warm and hospitable research environment This paper was finalized while I was participating in the program geometric aspects of nonlinear partial differential equations at Mittag-Leffler institute in Djursholm, Sweden during the fall of 2022.The research program is supported by Swedish Research Council grant no.2016-06596 2. Preliminaries 2.1.Notation.We define the monotone function J 1.3.Acknowledgements.The author warmly thanks Erik Lindgren for introducing the problem, proofreading this paper, for his helpful comments, and long hours of fruitful discussions.The author has partially been supported by the Swedish Research Council, grant no.2017-03736.During the development of this paper, I have been a PhD student at Uppsala University.p : R → R by