Hölder regularity for parabolic fractional p-Laplacian

Local Hölder regularity is established for certain weak solutions to a class of parabolic fractional p-Laplace equations with merely measurable kernels. The proof uses DeGiorgi’s iteration and refines DiBenedetto’s intrinsic scaling method. The control of a nonlocal integral of solutions in the reduction of oscillation plays a crucial role and entails delicate analysis in this intrinsic scaling scenario. Dispensing with any logarithmic estimate and any comparison principle, the proof is new even for the linear case.


INTRODUCTION
We are interested in the local Hölder regularity of weak solutions to a class of parabolic equations involving a fractional p-Laplacian type operator: (1.1) where E T = E × (0, T ] for some open set E ⊂ R N and some T > 0, and the nonlocal operator L is defined by (1.2) L u(x, t) = P.V.
Throughout this note, the parameters {s, p, N, C o , C 1 } are termed the data and we use γ as a generic positive constant in various estimates that can be determined by the data only.
The formal definition of weak solution to (1.1) -(1.3) and notations can be found in Section 1.2.We proceed to present our main result as follows.
Theorem 1.1.Let u be a locally bounded, local, weak solution to (1.1) - (1.3) in E T with p > 1.Then u is locally Hölder continuous in E T .More precisely, there exist constants γ > 1 and β ∈ (0, 1) that can be determined a priori only in terms of the data, such that for any 0 < ̺ < R < R, there holds In particular, a solution is allowed to grow at infinity.Whereas if u is globally bounded in R N × (0, T ), then ω can be taken as the global bound.This occurs if, for instance, proper initial/boundary data are prescribed, cf.[4,6].In addition, if u is globally bounded in R N × (−∞, T ), then u is, a fortiori, a constant by the oscillation estimate.
Remark 1.2.Theorem 1.1 continues to hold for more general structures.For instance, one can consider the kernel satisfying Also, proper lower order terms can be considered, cf.[8].However, we will concentrate on the actual novelty and leave possible generalizations to the motivated reader.
1.1.Novelty and Significance.The nonlocal elliptic operator L as in (1.2) with a kernel like (1.3), especially when p = 2, has been a classical topic in Probability, Potential theory, Harmonic Analysis, etc.In addition, nonlocal partial differential equations arise from continuum mechanics and phase transition, from population dynamics, and from optimal control and game theory.We refer to [5,14] for a source of motivations and applications.
Local regularity for the nonlocal elliptic operator with merely measurable kernels is well studied, cf.[8,12,13,17,22] -just to mention a few.In [12,13], localization techniques are developed in order to establish Hölder regularity and Harnack's inequality for the elliptic operator.A logarithmic estimate plays a key role in [12,13].Whereas [8] further improves these results to functions in certain DeGiorgi classes and the logarithmic estimate is dispensed with.
Our contribution lies in establishing Hölder regularity for the parabolic fractional p-Laplace type equation with merely measurable kernels for all p > 1.The approach is structural, in the sense that we dispense with any kind of comparison principle.More generally, we find that the Hölder regularity is in fact encoded in a family of energy estimates in Corollary 2.1 and tools like logarithmic estimates or exponential change of variables play no role in our arguments. 1As such, the arguments are new even for p = 2 and they hold the promise of a wider applicability, for instance, in Calculus of Variations.
Unlike the elliptic operator or the parabolic operator with p = 2, the local behavior of a solution to the parabolic p-Laplacian is markedly different: it has to be read in its own intrinsic geometry.This is the guiding idea in the local operator theory, cf.[7,9,10,11,25].In terms of oscillation estimates, this idea leads to the construction of geometric sequences {R i } and {ω i } connected by the intrinsic relation (1.4) ess osc The nonlocal theory developed here is no exception.However, the nonlocal character of (1.1) needs to be carefully handled in this intrinsic scaling scenario.A new component brought by the nonlocality of the operator is a proper control of the so-called tail -a nonlocal integral of the solution (see (1.9)).Precisely, we have In other words, the nonlocal tail is controlled by the local oscillation, if the intrinsic relation (1.4) is verified.The tail estimate (1.5) in turn allows us to reduce the oscillation in the next step, and so on.This induction procedure can be illustrated by The local regularity theory for the nonlocal parabolic problem (1.1) with p = 2 is still at its inception.We believe the techniques developed in this note are flexible enough and provide a handy toolkit that can be used to fruitfully attack more general nonlocal parabolic equations.
1.2.1.Function Spaces.For p > 1 and s ∈ (0, 1), we introduce the fractional Sobolev space W s,p (R N ) by which is endowed with the norm .
Similarly, the fractional Sobolev space W s,p (E) for a domain E ⊂ R N can be defined.Moreover, we denote These spaces admit imbedding into proper Lebesgue spaces; we collect some in Appendix A.

Notion of Weak Solution. A function
where for all non-negative testing functions A function u that is both a local weak sub-solution and a local weak super-solution to (1.1) -(1.3) is a local weak solution.
Remark 1.3.As we are developing a local theory, the function space in (1.8) can be taken smaller.Namely, the notion of W s,p o (K) can be replaced by functions ϕ(•, t) ∈ W s,p (R N ) with a compact support in K for a.e.t.
Remark 1.4.To ensure the convergence of the global integral in (1.7), it suffices to weaken the L ∞ norm appearing in the condition (1.6) by the L 1 norm.However, in deriving the energy estimate of Proposition 2.1, the condition (1.6) is needed already.1.2.3.Some Notations.Throughout this note, we will use K ̺ (x o ) to denote the ball of radius ̺ and center x o in R N , and the symbols to denote (backward) cylinders with the indicated positive parameters.When the context is unambiguous, we will omit the vertex (x o , t o ) from the symbols for simplicity.When θ = 1, it is also omitted.
A nonlocal integral of u -termed the tail of u -inevitably appears in the theory, which we define as (1.9) Tail u; Q(R, S) := ess sup For any Q(R, S) ⊂ E T , the finiteness of this tail is guaranteed by (1.6).

ENERGY ESTIMATES
This section is devoted to energy estimates satisfied by local weak sub(super)-solutions to (1.1) -(1.3).We first introduce, for any k ∈ R, the truncated functions In what follows, when we state "u is a sub(super)-solution..." and use " ± " or " ∓ " in what follows, we mean the sub-solution corresponds to the upper sign and the super-solution corresponds to the lower sign in the statement.
Here, we have denoted w = u − k for simplicity.
Proof.We will only deal with the case of sub-solution as the other case is similar.Using ϕ = w + ζ p as a testing function in the weak formulation modulo a proper time mollification (cf.Appendix B), the last terms on the right/left-hand side of the energy estimate are rather standard.We only treat the integral resulting from the fractional diffusion part, which, due to the support of ζ and symmetry of the integrand, can be split into two parts, that is, where we have used dµ = K(x, y, t)dydxdt for simplicity.
Let us manipulate the first integral, which is the leading term.To this end, we denote To proceed, we need an elementary inequality: (2.4) This simply follows from the mean value theorem and Young's inequality: We may apply (2.4) Now let us treat the second integral I 2 , which yields the only nonlocal integral in the energy estimate.Indeed, we first estimate As a result, we may estimate by the condition (1.3) on the kernel K, Note that the finiteness of the above nonlocal integral is guaranteed by (1.6).This term will evolve into the tail term (1.9) in the forthcoming theory.
Finally, we can put all these estimates together and use the condition (1.3) on the kernel K to conclude.
The above energy estimate can be written in K R × (t o − S, t) for any t ∈ (t o − S, t o ).As usual, this will lead to an L ∞ estimate in the time variable on the left, due to the arbitrariness of t.Further, by choosing a proper cutoff function ζ, we derive the following two types of energy estimates from Proposition 2.1, which encode all the information needed to show Theorem 1.1.

Corollary 2.1. Let u be a local weak sub(super)-solution to
Here, we have denoted w = u − k for simplicity.

PRELIMINARY TOOLS
In this section, we collect the main modules of the proof of Hölder regularity.An important feature is that the tail term appears in these modules via an either-or form.This feature clarifies the role of the tail and greatly facilitates the delicate intrinsic scaling arguments to be unfolded in the next two sections.To streamline, we derive them from the energy estimates in Proposition 2.1.Nevertheless, it will be clear from their proofs that Corollary 2.1 actually suffices.We also stress that the arguments in this section are given in a unified fashion for all p > 1.
Throughout this section, let be a cylinder included in E T .We introduce numbers µ ± and ω satisfying The first result concerns a DeGiorgi type lemma.For simplicity, we omit the vertex For some δ, ξ ∈ (0, 1) set θ = δ(ξω) 2−p and assume Q ̺ (θ) ⊂ Q.There exists a constant ν ∈ (0, 1) depending only on the data and δ, such that if . Moreover, we have the dependence ν ≈ δ q for some q > 1 depending on p and N .
Proof.It suffices to show the case of super-solution with µ − = 0. Assume (x o , t o ) = (0, 0) and define for n Recalling w − = (u − k n ) − , we treat the three terms on the right-hand side of the energy estimate as follows.For the first term, we estimate where we have defined For the second term, we observe that for |y| ≥ ̺ n and |x| ≤ ̺n , there holds consequently, recalling also u ≥ µ − = 0 a.e. in Q by assumption, we estimate In the last line, we have enforced For the third term, it is quite standard to obtain Collecting these estimates on the right-hand side of the energy estimate, we arrive at Now set 0 ≤ φ ≤ 1 to be a cutoff function in Q n , which vanishes outside Q n , equals the identity in Q n+1 and satisfies |Dφ| ≤ 2 n /̺.An application of the Hölder inequality and the Sobolev imbedding (cf.Proposition A.3 with d = 2 −n−4 ) gives that To obtain the last line, we used the triangle inequality for some c = c(p), such that Plugging this into the second-to-last line and employing the above energy estimate, the last line follows.Notice also from Proposition A.3 there holds In terms of Y n = |A n |/|Q n |, this estimate leads to the recursive inequality Proof.Assume (x o , t o ) = (0, 0).It suffices to show the case of super-solutions with µ − = 0. Let us first examine the energy estimate of Proposition 2.1 in Q(R, S) ≡ K ̺ × (0, θ̺ sp ) for some θ to be determined.Note that the time level t o − S in Proposition 2.1 corresponds to t = 0 here.Let ζ(x) be a time independent, piecewise smooth, cutoff function in K ̺ that vanishes on ∂K ̺ .If we take the level k ≤ ξω, the spatial integral at t = 0 (i.e. the term at the time level t o − S in Proposition 2.1) vanishes due to the assumption that u( Note that while shrinking the base balls K n , K n , K n and K n along ̺ n , the height of the cylinders is fixed.For the piecewise smooth function ζ(x) in K n , we choose it to vanish outside K n , be equal to 1 in K n , and satisfy |Dζ| ≤ 2 n /̺.By a similar treatment of the right-hand side as in Lemma 3.1, after enforcing ̺ R we may obtain that ess sup where Then we may proceed as in Lemma 3.1 to obtain the recursive inequality then Y n → 0. To finish the proof, we choose θ = ν o (ξω) 2−p .
The following lemma propagates measure theoretical information forward in time.
Lemma 3.3.Let u be a locally bounded, local weak sub(super)-solution to (1.1) Introduce parameters ξ and α in (0, 1).There exist δ and ε in (0, 1), depending only on the data and α, such that if provided this cylinder is included in Q.Moreover, we may trace the dependences by ε ≈ α and δ ≈ α p+1 .
The following measure shrinking lemma is usually a delicate part in the theory of the parabolic p-Laplacian.However, the term that involves mixed positive/negative truncations in the energy estimate greatly simplifies the argument.For ease of notation, we omit the vertex Lemma 3.4.Let u be a locally bounded, local weak sub(super)-solution to (1.1) -(1.3) in E T .Suppose that for some δ, σ and ξ in (0, 1  2 ), there holds There exists γ > 0 depending only on the data, such that either Proof.It suffices to show the case of super-solutions with µ − = 0.For simplicity, we assume (x o , t o ) = (0, 0).We employ the energy estimate of Proposition 2.1 in K 2̺ × (−θ̺ p , 0] with the truncation w − = (u − k) − and k = σξω, and introduce a cutoff function ζ in K 2̺ (independent of t) that is equal to 1 in K ̺ and vanishes outside K 3 2 ̺ , such that |Dζ| ≤ ̺ −1 .Then, we obtain from Proposition 2.1 that ¨Q̺(θ) w − (y, t) dydt The various terms on the right-hand side are estimated as in Lemma 3.1.Indeed, after enforcing ̺ R sp p−1 Tail(u − ; Q) ≤ σξω, they are bounded by γ (σξω) p δ̺ sp |Q ̺ (θ)|.The left-hand side is estimated by extending the integrals over smaller sets and by using the given measure theoretical information: Combining these estimates and properly adjusting relevant constants, we conclude the proof.) in E T .Suppose for some constants α, ξ ∈ (0, 1), there holds Then there exist constants δ, η ∈ (0, 1) depending only on the data and α, such that either ̺ R Moreover, we have δ ≈ α p+1 and η ≈ α q for some q > 1 depending on the data.
Proof.Assuming (x o , t o ) = (0, 0) and µ − = 0 for simplicity, it suffices to deal with supersolutions.Rewriting the measure theoretical information at the initial time t o = 0 in the larger ball K 4̺ and replacing α by and apply Lemma 3.3 to obtain δ, ε ∈ (0, 1), such that This measure theoretical information for each slice of the time interval in turn allows us to apply Lemma 3.4 with ξ and α there replaced by εξ and 1 2 4 −N α, since σ ∈ (0, 1) and δ(σεξω) 2−p ≤ δ(ξω) 2−p .Note that the above inequality used the fact that p ≤ 2. Letting ν be determined in Lemma 3.1 in terms of the data and δ, we further choose σ according to Lemma 3.4 to satisfy such a choice of σ permits us to apply Lemma 3.1 and conclude that u ≥ 1 2 σεξω a.e. in K 2̺ × δ(ξω) 2−p (4̺) sp − δ(σεξω) 2−p (2̺) sp , δ(ξω) 2−p (4̺) sp .The proof is completed by defining η = σε and properly adjusting relevant constants in dependence of the data.
Based on Proposition 4.1, the remaining part is devoted to the proof of Theorem 1.1 for 1 < p ≤ 2. All constants determined in the course of the proof are stable as p → 2.
Let us suppose the first alternative holds for instance.An appeal to Proposition 4.1 with α = 1 2 , ξ = 1 4 and ̺ = cR determines η ∈ (0, 1 2 ) and yields that either (4.2) ω) 2−p ), which, thanks to (4.1), gives the reduction of oscillation (4.3) ess osc The number c is chosen to ensure that (4.2) does not happen.Indeed, we may first estimate This can be seen by the definitions of ω and the tail, Then, using (4.4) we choose (4.5) such that (4.2) does not occur.Note that (4.5) is not the final choice of c yet and it is subject to a further smallness requirement.
Next we set R 1 = λR for some λ ≤ c to verify the set inclusion As a result of this inclusion and (4.3) we obtain which plays the role of (4.1) in the next stage.

4.3.
The Induction.Now we may proceed by induction.Suppose up to i = 0, 1, The induction argument will show that the above oscillation estimate continues to hold for the (j + 1)-th step.Let δ be fixed as before, whereas c ∈ (0, 1) is subject to a further choice.To reduce the oscillation in the next stage, we basically repeat what has been done in the first step, now with µ ± j , ω j , R j , Q j , etc.In fact, we define τ := δ( 1 4 ω j ) 2−p (cR j ) sp and consider two alternatives Like in the first step, we may assume µ + j −µ − j ≥ 1 2 ω j , so that one of the two alternatives must hold.Otherwise the case µ + j − µ − j < 1 2 ω j can be trivially incorporated into the forthcoming oscillation estimate (4.8).
Let us suppose the first case holds for instance.An application of Proposition 4.1 in Q j , with α = 1 2 , ξ = 1 4 and ̺ = cR j yields (for the same η as before) that either (4.7) ), which, thanks to the j-th induction assumption, gives the reduction of oscillation (4.8) ess osc The final choice of c is made to ensure that (4.7) does not happen, independent of j.This hinges upon the following tail estimate (4.9) Tail u − µ − j − ; Q j ≤ γω j .To prove this, we first rewrite the tail as follows: The first integral is estimated by using the definition of ω.Namely, for any t ∈ (−ω 2−p j R sp j , 0), Whereas the second integral is estimated by using the simple fact that, for i = 1, 2, Namely, for any t ∈ (−ω 2−p j R sp j , 0), Combine them and further estimate the tail by The summation in the last display is bounded by (1 − η) 1−p if we restrict the choice of λ by Consequently, the tail estimate (4.9) is proven and (4.7) does not happen, if we choose c to verify (4.10) The final choice of c is made out of the smaller one of (4.5) and (4.10).
Let R j+1 = λR j for some λ ∈ (0, 1) to verify the set inclusion Note that the choice of λ in the last display may have been adjusted from the precious one in (4.6) due to the possible change of c made in (4.10).The final choice of λ is As a result of the inclusion (4.11) and (4.8) we obtain which completes the induction argument.From now on, the deduction of a Hölder modulus of continuity becomes quite standard; cf.[9, Chapter III, Proposition 3.1].

PROOF OF THEOREM 1.1: p > 2
As in the previous section, we first introduce ω) 2−p and some a ∈ (0, 1) to be determined.By properly shrinking R, we may assume that Q o ⊂ Q R and set Without loss of generality, we take (x o , t o ) = (0, 0).Then the following intrinsic relation holds: (5.1) As before, the choice of Q R is made to ensure (5.1), on which the subsequent arguments are based.
Unlike the case 1 < p ≤ 2, an expansion of positivity for the case p > 2 requires addition technical complication.To deal with this case, we instead refine DiBenedetto's argument in [10]: On one hand, the tail needs a great care in this intrinsic scaling scenario; on the other hand, we dispense with any kind of logarithmic estimate and just rely on the energy estimate.5.1.The First Alternative.In this section, we work with u as a super-solution near its infimum.Furthermore, we assume (5.2) 2 ω, will be considered later.Suppose a, c ∈ (0, 1) verify that a > 2c sp for the moment (which will be confirmed in (5.11) and (5.12)), and for some t ∈ − aθR sp + θ(cR) sp , 0 , there holds , where ν is the constant determined in Lemma 3.1 (with δ = 1) in terms of the data.According to Lemma 3.1 with δ = 1, ξ = 1 4 and ̺ = cR, we have either To proceed, we restrict c so that (5.4) does not happen.Indeed, since, according to the definition of ω, the tail can be easily estimated by (cf.(4.4)) Note that this is not the final choice of c yet and it is subject to further smallness requirements in the course of the proof.The pointwise estimate in (5.5) at t * = t − θ( 1 2 cR) sp allows us to apply Lemma 3.2 with ̺ = 1 2 cR and obtain that for some ξ o ∈ (0, 1 8 ), either (5.7) In this way, the estimate (5.8) can be claimed up to t = 0 and yields the reduction of oscillation (5.10) ess osc Note that in the dependence of ξ o , the constants a and c are still to be determined.For the moment, let us suppose c has been fixed.We then select a to ensure (5.7) does not occur.In fact, since the tail is bounded by γω as before, we impose where we have employed the selection of ξ o in (5.9).Consequently, the above display yields the relation of a and c, that is, Hence, by this choice, the estimate (5.7) does not occur and the reduction of oscillation (5.10) actually holds.Moreover, the assumption a > 2c sp made at the beginning is verified, if we use (5.11) and further restrict c by (5.12) The number a will be eventually fixed via (5.11), once we determine c in the end.
5.2.The Second Alternative.In this section, we work with u as a sub-solution near its supremum.Suppose (5.3) does not hold for any t ∈ − aθR sp + θ(cR) sp , 0 .Due to (5.2), this can be rephrased as Based on this, it is not hard to find some t * ∈ t − θ(cR) sp , t − 1 2 νθ(cR) sp , such that Indeed, if the above inequality were not to hold for any s in the given interval, then which would yield a contradiction.
Starting from this measure theoretical information, we may apply Lemma 3.3 (with α = 1 2 ν and ̺ = cR) to obtain δ and ε depending on the data and ν, such that, for some ξ 1 ∈ (0, 1  4 ), either (5.13) The number ξ 1 is chosen to satisfy In this way, the measure theoretical information (5.14) can be claimed up to the time level t.Whereas the constant c is again chosen so small that (5.13) does not happen.A simple calculation as before gives Consequently, the measure theoretical information (5.14) yields (5.16) |K cR | for all t ∈ − aθR sp + θ(cR) sp , 0 , thanks to the arbitrariness of t.Given (5.16), we want to apply Lemma 3.4 with δ = 1, ξ = εξ 1 and ̺ = cR next.To this end, we first let ν be fixed in Lemma 3.1 (with δ = 1 ) and choose σ ∈ (0, 1  2 ) to satisfy Then we use (5.11) and restrict c further to satisfy (5.17) In this way, the measure theoretical information (5.16) gives that which allows us to implement Lemma 3.4.Namely, there exists γ > 0 depending only on the data, such that either (5.18) A key step lies in determining c as done in (5.6), (5.15), (5.12), (5.17) and (5.20), such that the alternative involving the tail, along the course of the arguments, does not really occur and hence (5.22) can be reached.This hinges upon the following estimate of the tail: The computations leading to the above tail estimate can be performed as those leading to (4.9).We omit the details to avoid repetition.After the number c is determined independent of j, the number a is finally chosen via the relation (5.11).Hence the induction is completed and the derivation of a Hölder modulus of continuity follows.

APPENDIX A. FRACTIONAL SOBOLEV INEQUALITIES
Note that the definition of the space W s,p in Section 1.2 is also valid for p = 1, though we will not use it.An elementary proof of the following result can be retrieved from [14,Theorem 6.5].It is our intention to circumvent a more advanced theory of function spaces, extension domains, etc. and keep this section as elementary as possible.
The following local version is a direct consequence of Proposition A.1.Here and in the sequel, we omit the reference to the center of a ball K R .In the last line, we used the fact that y ∈ K 1−d (recalling supp u) and x ∈ R N \ K 1 .Consequently, the desired inequality for any κ in the interval follows from an application of Hölder's inequality.Note also the constant C is actually independent of κ in this case.Next, we consider the case sp ≥ N .A first observation is that W s,p (K 1 ) ⊂ W s,p (K 1 ) for any 0 < s ≤ s < 1. Quantitatively, we estimate κp N , which verifies sp < N , and apply (A.1) in W s,p (K 1 ), together with the above observation, to conclude.
Based on Proposition A.2, the following parabolic imbedding is in order.It plays an essential role in proving DeGiorgi type lemmas.
Proposition A.3.Let s ∈ (0, 1), p ≥ 1 and For any function which is compactly supported in K (1−d)R for some d ∈ (0, 1) and for a.e.t ∈ (t 1 , t 2 ), there holds ˆt2 for some positive constant C = C(s, p, N ), where Proof.The inequality is scaling invariant in R. Hence, it suffices to show it for R = 1.An application of Hölder's inequality, followed by Proposition A.2 and the choice of κ, yields that This finishes the proof.

APPENDIX B. TIME MOLLIFICATION
The time derivative of a weak solution in general does not exist in the Sobolev sense.On the other hand, it is desirable to use the solution in a testing function when we derive the energy estimate in Proposition 2.1.To overcome this well-organized difficulty, we find the following mollifier quite convenient.Namely, we introduce for any v ∈ L 1 (E T ), Properties of this mollification can for instance be found in [20,Lemma 2.2].In particular, by direct differentiation, we obtain the following identities: They are paired by the identity Another fact, we will reply on is that, if v ∈ L q (E T ) for some q ≥ 1, then v h , v h ∈ L q (E T ), and moreover, as h → 0 we have Now we take on a rigorous justification of formal calculations in Proposition 2.1.For ease of notation, we denote On the right-hand side of the last display, the second term is discarded due to its non-negative contribution, whereas the third term converges to 0 as h → 0 owing to (B.2).For the first term, we integrate by parts in time, let h → 0 and then let ε → 0, to obtain Once this is shown, we will justify the formal calculations in the proof of Proposition 2.1.
For this purpose, we take the difference between the two integrals and split the obtained integral into two parts, that is,  To prove (B.4), the numerator of the integrand needs to be controlled.Using the triangle inequality and the fact that ζ and ψ ε are bounded by 1, we compute ϕ h (x, t) − ϕ h (y, t) p ≤ c u h(x, t) − u h(y, t) p + c u h(y, t)

, 1 Remark 1 . 1 .
provided the cylinders (x o , t o ) + Q R (ω 2−p ) ⊂ (x o , t o ) + Q R are included in E T , where ω = 2 ess sup (xo,to)+Q R |u| + Tail u; (x o , t o ) + Q R .Theorem 1.1 has been formulated independent of any initial/boundary data.While local, the oscillation estimate bears global information via the tail of u; see (1.9).

,
where Y n = |A n |/|Q n | and the constants γ, b depend only on the data.Hence, by [9, Chapter I, Lemma 4.1], there exists a positive constant ν o depending only on the data, such that if

4 . 2 4. 1 .Proposition 4 . 1 .
PROOF OF THEOREM 1.1: 1 < p ≤ Expansion of Positivity.Suppose the cylinder Q and the numbers µ ± and ω are defined as in Section 3. The key ingredient of the reduction of oscillation lies in the following expansion of positivity valid for 1 < p ≤ 2. Let u be a locally bounded, local, weak sub(super)-solution to (1.1) -(1.3

Proposition A. 2 .ˆKR |u| κp dx 1 κ
Let s ∈ (0, 1) and p ≥ 1.For any function u ∈ W s,p (K R ) that is compactly supported in K (1−d)R with some d ∈ (0, 1), there holds− ≤ CR sp ˆKR − ˆKR u(x) − u(y) p |x − y| N +sp dxdy + C d N +sp − ˆKR |u(x)| p dx for some positive constant C = C(κ, s, p, N ), where κ ∈ 1, N N −sp , sp < N, [1, ∞), sp ≥ N.Proof.It suffices to show the inequality for R = 1, thanks to its scaling invariance in R. Since u is compactly supported in K 1 , we may view it as a function in W s,p (R N ) upon zero extension.This fact is verified by a simple calculation (similar to what follows).First consider the case sp < N .According to Proposition A.1) − u(y) p |x − y| N +sp dxdy + 2C ˆK1 ˆRN \K 1 |u(y)| p |x − y| N +sp dxdy ≤ C ˆK1 ˆK1 u(x) − u(y) p |x − y| N +sp dxdy + C d N +sp ˆK1 |u(y)| p dy.

1 2 ˆKR u − k 2 + ζ p dx to to−S − 1 2k 2 +
¨Q(R,S)u − ∂ t ζ p dxdt.These are the last terms on the left/right-hand side of the energy estimate in Proposition 2.1.The next goal is to show that the second integral in (B.3) converges to ˆT 0 ˆRN ˆRN A • ϕ(x, t) − ϕ(y, t) dydxdt, if we send h → 0, where ϕ = u − k + ζ p ψ ε .