Parabolic L p Dirichlet Boundary Value Problem and VMO-type time-varying domains
aa r X i v : . [ m a t h . A P ] M a y PARABOLIC L p DIRICHLET BOUNDARY VALUE PROBLEMAND
VMO -TYPE TIME-VARYING DOMAINS
MARTIN DINDOŠ, LUKE DYER, AND SUKJUNG HWANG
Abstract.
We prove the solvability of the parabolic L p Dirichlet boundaryvalue problem for 1 < p ≤ ∞ for a PDE of the form u t = div( A ∇ u ) + B · ∇ u on time-varying domains where the coefficients A = [ a ij ( X, t )] and B = [ b i ]satisfy a certain natural small Carleson condition. This result brings the stateof affairs in the parabolic setting up to the elliptic standard.Furthermore, we establish that if the coefficients of the operator A, B satisfya vanishing Carleson condition and the time-varying domain is of VMO typethen the parabolic L p Dirichlet boundary value problem is solvable for all1 < p ≤ ∞ . This result is related to results in papers by Maz’ya, Mitrea andShaposhnikova, and Hofmann, Mitrea and Taylor where the fact that boundaryof domain has normal in VMO or near VMO implies invertibility of certainboundary operators in L p for all 1 < p ≤ ∞ which then (using the method oflayer potentials) implies solvability of the L p boundary value problem in thesame range for certain elliptic PDEs.Our result does not use the method of layer potentials, since the coefficientswe consider are too rough to use this technique but remarkably we recover L p solvability in the full range of p ’s as the two papers mentioned above. Introduction
Let us consider a parabolic differential equation on a time-varying domain Ω ofthe form ( u t = div( A ∇ u ) + B · ∇ u in Ω ⊂ R n +1 ,u = f on ∂ Ω , (1.1)where A = [ a ij ( X, t )] is a n × n matrix satisfying the uniform ellipticity conditionwith X ∈ R n , t ∈ R . That is, there exists positive constants λ and Λ such that λ | ξ | ≤ X i,j a ij ( X, t ) ξ i ξ j ≤ Λ | ξ | (1.2)for almost every ( X, t ) ∈ Ω and all ξ ∈ R n . In addition, we assume that thecoefficients of A and B satisfy a natural, minimal smoothness condition, (1.6), andwe do not assume any symmetry on A .It has been observed via the method of layer potentials that when the domain onwhich we consider certain boundary value problems for elliptic or parabolic PDEsis sufficiently smooth the question of L p invertibility of certain boundary operatorcan be resolved using the Fredholm theory since this operator is just a compactperturbation of the identity. This observation then implies invertibility of thisboundary operator for all 1 < p ≤ ∞ and hence solvability of the corresponding L p boundary value problem in this range.The notion of how smooth the domain has to be for the above observation to holdhas evolved. Initial results for constant coefficient elliptic PDEs required domains Luke Dyer was supported by The Maxwell Institute Graduate School in Analysis and its Ap-plications, a Centre for Doctoral Training funded by the UK Engineering and Physical SciencesResearch Council (grant EP/L016508/01), the Scottish Funding Council, the University of Edin-burgh, and Heriot-Watt University. of at least C ,α type. This was reduced to C domains in an important paperof Fabes, Jodeit and Rivière [FJR78]. Later the method of layer potentials wasadapted to variable coefficient settings and the results were extended to ellipticPDEs with variable coefficients [Din08] on C domains.Further progress was made after advancements in singular integrals theory onsets that are not necessary of graph-type [Sem91, HMT10]. It turns out that com-pactness of the mentioned boundary operator only requires that the normal (whichmust be well defined at almost every boundary point) belongs to VMO.This observation for the Stokes system was made in [MMS09] where boundaryvalue problems for domains whose normal belongs to VMO (or is near to VMOin the BMO norm) were considered. In [HMT15] symbol calculus for operators oflayer potential type on surfaces with VMO normals was developed and applied tovarious elliptic PDEs including elliptic systems.So far we have only mentioned elliptic results. One of the first results for theheat equation in Lipschitz cylinders is by Brown [Bro89]. Here the domain con-sidered is time independent and Fourier methods in the time variable are used.Domains of time-varying type for the heat operator were first considered the pa-pers [LM95, HL96] and again the method of layer potentials was used to establish L solvability. The question of solvability of various boundary value problems forparabolic PDEs on time-varying, domains has long history. Recall, that in theelliptic setting [Dah77] has shown in a Lipschitz domain that the harmonic meas-ure and surface measure are mutually absolutely continuous, and that the ellipticDirichlet problem is solvable with data in L with respect to surface measure. R.Hunt then asked whether Dalhberg’s result held for the heat equation in domainswhose boundaries are given locally as functions φ ( x, t ), Lipschitz in the spatial vari-able. It was conjectured (due to the natural parabolic scaling) that the correctregularity of φ ( x, t ) should be a Hölder condition of order 1 / t and Lipschitz in x . It turns out that under this assumption the parabolic measureassociated with the equation (1.1) is doubling [Nys97].However, in order to answer R. Hunt’s question positively one has to considermore regular classes of domains than the one just described above. This follows fromthe counterexample of [KW88] where it was shown that under just the Lip(1 , / ∂ t − ∆) might not be mutually absolutely continuous withthe natural surface measure. The issue was resolved in [LM95] where it was estab-lished that mutual absolute continuity of caloric measure and a certain parabolicanalogue of the surface measure holds when φ has 1 / R n ) space, which is a slightly stronger condition than Lip(1 , / L p Dirichlet problem was solvable for all p > p ′ for some potentially very large p ′ (dueto the technique used there is no control on the size of p ′ ). Finally [DH16] hasestablished L p solvability 2 ≤ p ≤ ∞ in domains that are locally of Lewis-Murraytype under a small Carleson condition.While researching literature on domains of Lewis-Murray type and ways thisconcept can be localized (in the time variable the half-derivative is a nonlocaloperator and hence any condition imposed on it is difficult to localize) we haverealized that important results we have planned to rely on have issues (either intheir proofs or even worse are simply false, see in particular remark 2.7 in thenext section). This has prompted us to write section 2.1 on parabolic domains in ARABOLIC L p DIRICHLET PROBLEM AND VMO-TYPE DOMAINS 3 substantially more detail we originally intended to. This sets the literature recordstraight and more importantly in detail explains the concept of localized domainsof Lewis-Murray type. For readability of the paper and this section we have movedlong proofs into an appendix.In this paper we establish L p solvability results for parabolic PDEs on time-varying cylinders satisfying locally the Lewis-Murray condition in the full range1 < p ≤ ∞ improving the solvability range from [DH16] as well as older resultssuch as [HL96], where only p = 2 was considered. The coefficients we consider arevery rough and in particular the method of layer potentials cannot be used. Despitethis we recover (in the parabolic setting) an analogue of [MMS09] and [HMT15].When the domain Ω, on which the parabolic PDE is considered, is of VMO type(that is certain derivatives both in temporal and spatial variables will be in VMO)and the coefficients of the operator satisfy a vanishing Carleson condition the L p solvability can be established for all 1 < p ≤ ∞ . Remarkably this is the full rangeof solvability that holds for smooth coefficients (via the layer potential method).Our proof is however completely different from the layer potential method, forexample at no point is compactness used. The proof is also substantially differentthan the case 2 ≤ p ≤ ∞ of [DH16] in the following way. We were inspiredby [DPP07] and have used a so called p -adapted square function in order to prove L p solvability. However, due to the presence of parabolic term a second squarefunction type object will arise, namely ˆ Ω | u t ( X, t ) | | u ( X, t ) | p − δ ( X, t ) d X d t, (1.3)where δ ( X, t ) is the parabolic distance to the boundary. When p = 2 such object wascalled the “area function” and in [DH16] it was shown that it can be dominatedby the usual square function. It turns out however that the case 1 < p < u after a substantial effort.There is also an issue of whether the p -adapted square function is actually well-defined and locally finite (as the exponent on | u | is negative). We prove that when u is a solution of a parabolic PDE the p -adapted square function is indeed welldefined by adapting a recent regularity result [DP16]. The paper [DP16] deals withcomplex coefficient elliptic PDEs but the method used there can be adapted to theparabolic setting; see Theorem 4.1 for details.Many results in the parabolic setting, are motivated by previous results in theelliptic setting and ours is not different. Let us therefore overview the major ellipticresults related to our main theorem.The papers [KKPT00] and [KP01] started the study of non-symmetric divergenceelliptic operators with bounded and measurable coefficients. [KP01] used [KKPT00]to show that the elliptic measure of operators satisfying a type of Carleson meas-ure condition is in A ∞ and hence the L p Dirichlet problem is solvable for some,potentially large, p . In [DPP07], the authors improved the result of [KP01] in thefollowing way. They showed that if δ ( X ) − (cid:18) osc B δ ( X ) / ( X ) a ij (cid:19) and δ ( X ) sup B δ ( X ) / ( X ) b i ! (1.4)are densities of Carleson measures with vanishing Carleson norms then the L p Dirichlet problem is solvable for all 1 < p ≤ ∞ . A similar result for the ellipticNeumann and regularity boundary value problem was established in [DPR17].
MARTIN DINDOŠ, LUKE DYER, AND SUKJUNG HWANG
The parabolic analogue of the elliptic Carleson condition (1.4) is that δ ( X, t ) − sup i,j (cid:18) osc B δ ( X,t ) / ( X,t ) a ij (cid:19) + δ ( X, t ) sup B δ ( X,t ) / ( X,t ) b i ! (1.5)is the density of a Carleson measure on Ω with a small Carleson norm and δ ( X, t )is the parabolic distance of a point (
X, t ) to the boundary ∂ Ω.The condition (1.5) arises naturally as follows. Let Ω = { ( x , x, t ) : x > φ ( x, t ) } for a function φ which satisfies the Lewis-Murray condition above. Let ρ : U → Ωbe a mapping from the upper half space U to Ω. Consider v = u ◦ ρ . It will followthat if u solves (1.1) in Ω then v will be a solution to a parabolic PDE similar to (1.1)in U . In particular if ρ is chosen to be the mapping in (2.26) then the coefficientsof the new PDE for v will satisfy a Carleson condition like (1.5), c.f. Lemma 2.18,provided the original coefficients (for u ) were either smooth or constant.Furthermore, if we do not insist on control over the size of the Carleson normthen we can still infer solvability of the L p Dirichlet problem for large p , as in [HL01,Riv03, Riv14].Finally, we ready to state our main result; some notions used here are defined indetail in section 2. Theorem 1.1.
Let Ω be a domain as in definition 2.10 with character ( ℓ, η, N, d ) and let A be bounded and elliptic (1.2) , and B be measurable. Consider any
such that if for some r > { η, k µ k C,r } < K the L p Dirichlet boundary value problem (1.1) is solvable (c.f. definition 2.26).Moreover, the following estimate holds for all continuous boundary data f ∈ C ( ∂ Ω) k N ( u ) k L p ( ∂ Ω , d σ ) . k f k L p ( ∂ Ω , d σ ) , where the implied constant depends only on the operator, n , p and character ( ℓ, η, N, d ) ,and N ( u ) is the non-tangential maximal function of u . Corollary 1.2.
In particular, if Ω is of VMO-type ( η in the character ( ℓ, η, N, d ) can be taken arbitrary small), and the Carleson measure µ from Theorem 1.1 is avanishing Carleson measure then the L p Dirichlet boundary value problem (1.1) issolvable for all < p ≤ ∞ . Preliminaries
Here and throughout we consistently use ∇ u to denote the gradient in the spatialvariables and u t or ∂ t u the gradient in the time variable. ARABOLIC L p DIRICHLET PROBLEM AND VMO-TYPE DOMAINS 5
Parabolic Domains.
In this subsection we define a class of time-varyingdomains whose boundaries are given locally as functions φ ( x, t ), Lipschitz in thespatial variable and satisfying the Lewis-Murray condition in the time variable. Ateach time τ ∈ R the set of points in Ω with fixed time t = τ , that is Ω τ = Ω ∩{ t = τ } ,is a non-empty bounded Lipschitz domain in R n . We start with a discussion of theLewis-Murray condition, give a summary and clarification of the results in theliterature, and introduce some new equivalent definitions.We define a parabolic cube in R n − × R , for a constant r >
0, as Q r ( x, t ) , = { ( y, s ) ∈ R n − × R : | x i − y i | < r for all 1 ≤ i ≤ n − , | t − s | / < r } . Let J r ⊂ R n − be a spatial cube of radius r . For a given f : R n → R let f Q r = 1 | Q r | ˆ Q r f ( x, t ) d x d t. When we write f ∈ BMO( R n ) we mean that f belongs to the parabolic version ofthe usual BMO space with the norm k f k ∗ where k f k ∗ = sup Q r | Q r | ˆ Q r | f − f Q r | d x d t < ∞ . (2.1)Recall that the Lewis-Murray condition imposed that a half derivative in time of φ ( x, t ) belongs to parabolic BMO. There are a few different ways one can definehalf derivatives and BMO-Sobolev spaces and there are also some erroneous resultsin the literature which we correct here. To bring clarity, we start by discussing thevarious definitions in the global setting of a graph domain Ω = { ( x , x, t ) : x >φ ( x, t ) } , where φ : R n − × R → R . We follow the standard notation of [HL96].If g ∈ C ∞ ( R ) and 0 < α < D α are defined on the Fourier side by d D α g ( τ ) = | τ | α ˆ g ( τ ) . If 0 < α < D α g ( t ) = c ˆ R g ( t ) − g ( s ) | t − s | α d s. Therefore, we define the pointwise half derivative in time of φ : R n − × R → R tobe D t / φ ( x, t ) = c n ˆ R φ ( x, s ) − φ ( x, t ) | s − t | / d s, (2.2)for a properly chosen constant c n (c.f. [HL96]).However, this definition ignores the spatial coordinates. Instead by following [FR67]we may define the parabolic half derivative in time of φ : R n − × R → R to be d D n φ ( ξ, τ ) = τ k ( ξ, τ ) k ˆ φ ( ξ, τ ) , (2.3)where ξ and τ denote the spatial and temporal variables on the Fourier side respect-ively, and k ( x, t ) k = | x | + | t | / denotes the parabolic norm. In addition we definethe parabolic derivative (in space and time) of φ : R n − × R → R to be c D φ ( ξ, τ ) = k ( ξ, τ ) k ˆ φ ( ξ, τ ) . (2.4) D − is the parabolic Riesz potential. One can also represent D as D = n X j =1 R j D j , (2.5) MARTIN DINDOŠ, LUKE DYER, AND SUKJUNG HWANG where D j = ∂ j for 1 ≤ j ≤ n − D n is defined above and R j are the parabolicRiesz transforms defined on the Fourier side as c R j ( ξ, τ ) = iξ j k ( ξ, τ ) k for 1 ≤ j ≤ n − c R n ( ξ, τ ) = τ k ( ξ, τ ) k . (2.6)Furthermore the kernels of R j have average zero on (parabolically weighted) spheresaround the origin, obey the standard Calderòn-Zygmund kernel and therefore bystandard Calderòn-Zygmund theory each R j defines a bounded operator on L p ( R n )for 1 < p < ∞ and is bounded on BMO( R n ) [Pee66, FR66, FR67, HL96].We say that φ : R n − × R → R is Lip(1 , /
2) with Lipschitz constant ℓ if φ isLipschitz in the spatial variables and Hölder continuous of order 1 / | φ j ( x, t ) − φ j ( y, t ) | ≤ ℓ (cid:16) | x − y | + | t − s | / (cid:17) . (2.7)The Lewis-Murray condition on the domain Ω, for which they proved the mutualabsolute continuity of the caloric measure and the natural surface measure, is φ ∈ Lip(1 , /
2) and k D t / φ k ∗ ≤ η ; note this BMO norm is taken over R n .It is worth remarking that neither the operators D t / , D n or D easily lend them-selves to being localised to a function φ : Q d → R due to their non-local natures.However, our goal is provide a theory where the domain is locally given by graphswhich satisfy the Lewis-Murray condition. The parabolic nature of the PDE (espe-cially time irreversibility and exponential decay of solutions with vanishing bound-ary data) suggest we should expect to need only local conditions on the functionsdescribing the boundary.To this end we state the following theorems where we show some equivalentstatements to the Lewis-Murray condition for a global function φ : R n − × R → R .Furthermore, the final conditions admit themselves to both being localised easilyas well as amiable to extension, see Theorem 2.8 later for details on a extension.The equivalence of conditions (1) and (2) below is shown in [HL96] with anequivalence of norms in the small and large sense, see [HL96, (2.10) and Theorem7.4] for precise details, c.f. (2.5) and (2.6). Theorem 2.1.
Let φ : R n − × R → R and φ ∈ Lip(1 , / then the followingconditions are equivalent:(1) D t / φ ∈ BMO( R n ) (2) D n φ ∈ BMO( R n ) (3) D φ ∈ BMO( R n ) .Furthermore D n φ = R n D φ and so k D n φ k ∗ . k D φ k ∗ . We now extend this theorem by adding three more equivalent statements. Tomotivate condition (6) of Theorem 2.3 below we first recall a characterisation ofBMO from [Str80, p. 546]. Let M ( f, Q ) = | Q | ´ Q f denote the average of f over acube Q , and let ˜ Q ρ ( x ) be the cube of radius ρ with x in the upper right corner. Lemma 2.2 ([Str80]) . f ∈ BMO( R n ) is equivalent to sup Q r n X k =1 | Q r | ˆ Q r ˆ r (cid:12)(cid:12) M ( f, ˜ Q ρ ( x )) − M ( f, ˜ Q ρ ( x − ρe k )) (cid:12)(cid:12) d ρρ d x = B < ∞ , (2.8) where e k are the usual unit vectors in R n , and k f k ∗ ∼ B . The equivalence of conditions (3) and (4) in theorem below is a generalisationof [Str80] to the parabolic setting that is stated in [Riv03], c.f. [FS72, CT75, CT77].
ARABOLIC L p DIRICHLET PROBLEM AND VMO-TYPE DOMAINS 7
We have some question-marks over the proof given in [Riv03]; however the argumentwe give for condition (5) also works for condition (4) and hence the claim in [Riv03]is correct.
Theorem 2.3.
Let φ : R n − × R → R and φ ∈ Lip(1 , / then the followingconditions are equivalent:(3) D φ ∈ BMO( R n ) (4) sup Q r | Q r | ˆ Q r ˆ k ( y,s ) k≤ r | φ ( x + y, t + s ) − φ ( x, t ) + φ ( x − y, t − s ) | k ( y, s ) k n +3 d y d s d x d t = B (4) < ∞ , (2.9) (5) (a) sup Q r | Q r | ˆ Q r ˆ | y | Condition (6.a) doesn’t immediately look too similar to its supposedmotivation, (2.8) in Lemma 2.2. However, if we move back into Cartesian coordin-ates and undo the mean value theorem then we obtain something much similar toa combination of (2.8) and a endpoint version of [Str80, (3.1)] . The reason whywe can obtain the endpoint, whereas [Str80, (3.1)] can only be used for a fractionalderivative smaller than , is due to extra integrability and cancellation coming from (8.1) . Consider MARTIN DINDOŠ, LUKE DYER, AND SUKJUNG HWANG A ′ k = M (cid:0) φ, ˜ Q k ( y,s ) k ( x + y, t ) (cid:1) − M (cid:0) φ, ˜ Q k ( y,s ) k ( x, t ) (cid:1) − M (cid:0) φ, ˜ Q k ( y,s ) k ( x + y − k ( y, s ) k e k , t ) (cid:1) + M (cid:0) φ, ˜ Q k ( y,s ) k ( x − k ( y, s ) k e k , t ) (cid:1) ,A ′ n = M (cid:0) φ, ˜ Q k ( y,s ) k ( x + y, t ) (cid:1) − M (cid:0) φ, ˜ Q k ( y,s ) k ( x, t ) (cid:1) − M (cid:16) φ, ˜ Q k ( y,s ) k ( x + y, t − k ( y, s ) k ) (cid:17) + M (cid:16) φ, ˜ Q k ( y,s ) k ( x, t − k ( y, s ) k ) (cid:17) then condition (6.a) is equivalent to sup Q r n X k =1 | Q r | ˆ Q r ˆ k ( y,s ) k The statement ∇ φ ∈ BMO( R n − ) implies condition (5.a) follows from [Str80,Theorem 3.3]. In order to establish the second claim for the ease of notation letus fix Q r and k in 1 ≤ k ≤ n − 1. Then since | u ′ | ≤ y = x + λρu ′ ∈ Q r ) we get that (2.12) is boundedby ˆ ˆ S n − r ˆ | Q r | ˆ Q r (cid:12)(cid:12)(cid:0) M (cid:0) ∇ u, ˜ Q ρ ( y, t ) (cid:1) − M (cid:0) ∇ u, ˜ Q ρ ( y − ρe k , t ) (cid:1)(cid:1)(cid:12)(cid:12) d y d t d ρρ d u d λ. Then by Lemma 2.2 the two interior integrals are bounded by C k∇ φ k ∗ . There-fore (2.12) is controlled by C k∇ φ k ∗ . (cid:3) The opposite implications are likely to be false due the highly singular nature ofRiesz potentials, c.f. (2.5) and (2.6). Corollary 2.6. If k∇ φ k ∗ . η , and B (5 .b ) . η then k D φ k ∗ . η . Here we have replaced conditions (5.a) or (6.a) by slightly stronger but easier toverify condition k∇ φ k ∗ . η . We believe that, without too much extra work, onecould formulate our main theorem and associated lemmas with a local version ofcondition (5.a) in place of k∇ φ k ∗ . Remark 2.7. In [Riv03, Lemma 2.1] it is stated that another condition is equivalentto those given in Theorems 2.1 and 2.3; however this claim is not correct and onlyone of the stated implications holds.A result of Strichartz [Str80, Theorem 3.3] states that in the one dimensionalsetting D t / φ ( t ) ∈ BMO( R ) is equivalent to the one dimensional version of condi-tions (5.b) and (6.b) sup I ′ ⊂ R (cid:18) | I ′ | ˆ I ′ ˆ I ′ | φ ( t ) − φ ( s ) | | t − s | d t d s (cid:19) / ≤ B, (2.15) with B ∼ k D t / φ ( · ) k BMO( R ) .In [Riv03, Lemma 2.1] it is claimed that given φ : R n − × R → R and φ ∈ Lip(1 , / the pointwise n -dimensional analogue of (2.15)sup x ∈ R n − sup I ′ ⊂ R (cid:18) | I ′ | ˆ I ′ ˆ I ′ | φ ( x, t ) − φ ( x, s ) | | t − s | d t d s (cid:19) / ≤ B (2.16) ARABOLIC L p DIRICHLET PROBLEM AND VMO-TYPE DOMAINS 9 is equivalent to D φ ∈ BMO( R n ) with B ∼ k D φ k BMO( R n ) . This is incorrect. By [Str80] (2.16) is equivalent to D t / φ ( x, · ) ∈ BMO( R ) pointwise for a.e. x . After some tedious andtechnical calculations we were able to show sup x D t / φ ( x, · ) ∈ BMO( R ) implies D t / φ ∈ BMO( R n ) and hence D φ ∈ BMO( R n ) via condition (4) of Theorem 2.3.However, the converse is not true even if we assume more structure for the function D φ ( x, t ) . This is due to the fact that there is “no reasonable Fubini theorem relating BMO( R n ) to BMO( R ) ” [Str80, p. 558] .Fortunately the lack of a converse implication does not cast doubt over the sub-sequent results of [Riv03] since the author only uses the claimed eqivalence in thecorrect direction — that (2.16) implies D φ ∈ BMO( R n ) . Localisation. After the comprehensive review of the Lewis-Murray condition for agraph domain Ω we continue in our aim to construct a time-varying domain whichis locally described by local graphs φ j .For a vector x ∈ R n − we denote consider the norm | x | ∞ = sup i | x i | .Consider φ : Q d → R n − × R . The localised version of (2.11) from Theorem 2.3is simply sup Q r = J r × I r Q r ⊂ Q d | Q r | ˆ Q r ˆ I r | φ ( x, t ) − φ ( x, s ) | | t − s | d s d t d x < ∞ . (2.17)We write k f k ∗ ,d to be the BMO norm of f where the supremum in the BMOnorm, c.f. (2.1), is taken over all cubes Q r with r ≤ d . For a function f : J × I → R ,where J ⊂ R n − and I ⊂ R are closed bounded cubes we consider the norm k f k ∗ ,J × I defined as above where the supremum is taken over all parabolic cubes Q r contained in J × I . The norm k f k ∗ ,J × I,d is where the supremum is taken overall parabolic cubes Q r with r ≤ d contained in J × I . If the context is clear wesuppress the J × I and just write k f k ∗ or k f k ∗ ,d .Recall that VMO( R n ) is defined as the closure of all continuous functions inthe BMO norm or equivalently BMO functions f such that k f k ∗ ,d → d → d ( f, VMO) := inf h ∈ C k f − h k ∗ then f ∈ VMO if and only if d ( f, VMO) = 0; for f ∈ BMO this measures thedistance of f to VMO. In our case, the boundary of the parabolic domains weconsider can be locally described as a graph of a continuous function. However, asour domain is unbounded in time we may potentially require an infinite family oflocal graphs { φ j } . Therefore we need to measure the distance to VMO uniformlyacross this infinite family.Let δ : R + → R + , δ (0) = 0 and δ be continuous at 0 then we define C δ to be theset of continuous functions with the same modulus of continuity δ . That is C δ = { g ∈ C : | g ( x ) − g ( y ) | ≤ δ ( | x − y | ) for all x, y } . (2.18)Note that every family of equicontinuous functions can be represented as C δ forsome function δ and C = ∪ δ C δ . For f : Q d → R we define d ( f, C δ ) as d ( f, C δ ) = inf h ∈ C δ k f − h k ∗ ,Q d . We are now ready to state and prove result on extensibility of φ : Q d → R to aglobal function. Theorem 2.8. Let φ : Q d ⊂ R n − × R → R be Lip(1 , / with Lipschitz constant ℓ . If there exist a scale r , a constant η > and a modulus of continuity δ such that sup Q s = J s × I s Q s ⊂ Q d , s ≤ r | Q s | ˆ Q s ˆ I s | φ ( x, t ) − φ ( x, τ ) | | t − τ | d τ d t d x ≤ η (2.19) and d ( ∇ φ, C δ ) ≤ η (2.20) then there exists a scale d ′ ≤ d , that only depends on d , δ , η , and r and not φ ,such that for all Q r ⊂ Q d with r ≤ d ′ there exists a global Lip(1 , / function Φ : R n − × R → R with the following properties for all < ε < :(i) Φ | Q r = φ | Q r ,(ii) the Lip(1 , / constant of Φ is ℓ ,(iii) k∇ Φ k ∗ . ε η − ε + ηℓ , and(iv) sup Q s = J s × I s | Q s | ˆ Q s ˆ I s | Φ( x, t ) − Φ( x, τ ) | | t − τ | d τ d t d x . η .Therefore by Corollary 2.6, k D Φ k ∗ . ε η − ε + ηℓ . We again give the proof of this result in the appendix. We are now ready todefine the class of parabolic domains on which we will work. Motivated by theusual definition of a Lipschitz domain we have: Definition 2.9. Z ⊂ R n × R is an ℓ -cylinder of diameter d if there exists a coordin-ate system ( x , x, t ) ∈ R × R n − × R obtained from the original coordinate system bytranslation in spatial and time variables, and rotation only in the spatial variablessuch that Z = { ( x , x, t ) : | x | ≤ d, | t | / ≤ d, | x | ≤ ( ℓ + 1) d } and for s > s Z := { ( x , x, t ) : | x | < sd, | t | / ≤ sd, | x | ≤ ( ℓ + 1) sd } . Definition 2.10. Ω ⊂ R n × R is an admissible parabolic domain with character ( ℓ, η, N, d ) if there exists a positive scale r , and a modulus of continuity δ such thatfor any time τ ∈ R there are at most N ℓ -cylinders { Z j } Nj =1 of diameter d satisfyingthe following conditions:(1) ∂ Ω ∩ {| t − τ | ≤ d } = [ j ( Z j ∩ ∂ Ω) .(2) In the coordinate system ( x , x, t ) of the ℓ -cylinder Z j Z j ∩ Ω ⊃ (cid:8) ( x , x, t ) ∈ Ω : | x | < d, | t | < d , δ ( x , x, t ) ≤ d/ (cid:9) . (3) Z j ∩ ∂ Ω is the graph { x = φ j ( x, t ) } of a function φ j : Q d → R , with Q d ⊂ R n − × R , such that | φ j ( x, t ) − φ j ( y, s ) | ≤ ℓ (cid:16) | x − y | + | t − s | / (cid:17) and φ j (0 , 0) = 0 . (2.21) (4) d ( ∇ φ j , C δ ) ≤ η (2.22) and sup Q s = J s × I s Q s ⊂ Q d , s ≤ r | Q s | ˆ Q s ˆ I s | φ j ( x, t ) − φ j ( x, τ ) | | t − τ | d τ d t d x ≤ η . (2.23) Here and throughout δ ( x , x, t ) := dist (( x , x, t ) , ∂ Ω) and dist is the parabolic dis-tance dist[( X, t ) , ( Y, s )] = | X − Y | + | t − s | / .We say that Ω is of VMO type if η in the character ( ℓ, η, N, d ) can be takenarbitrarily small (at the expense of a potentially smaller d and r , and larger N ). ARABOLIC L p DIRICHLET PROBLEM AND VMO-TYPE DOMAINS 11 Remark 2.11. When (2.22) holds for small or vanishing η it follows that for afixed time τ the normal ν to the fixed-time spatial domain Ω τ = Ω ∩ { t = τ } can bewritten in local coordinates as ν = 1 | ( − , ∇ φ j ) | ( − , ∇ φ j ) and hence d ( ν, VMO) . η . Therefore Ω τ is similar to the domains considered inthe papers [MMS09] and [HMT15] which have dealt with the elliptic problems ondomains with normal in or near VMO . Remark 2.12. It follows from this definition that for each τ ∈ R the time-slice Ω τ of an admissible parabolic domain Ω ⊂ R n × R is a bounded Lipschitz domain in R n and they all have a uniformly bounded diameter. That is inf τ ∈ R diam(Ω τ ) ∼ d ∼ sup τ ∈ R diam(Ω τ ) , where d is the scale from definition 2.10 and the implied constants only dependson N . In particular, if O ⊂ R n is a bounded Lipschitz domain then the paraboliccylinder Ω = O × R is an example of a domain satisfying definition 2.10. Definition 2.13. Let Ω ⊂ R n × R be an admissible parabolic domain with character ( ℓ, η, N, d ) . The measure σ defined on sets A ⊂ ∂ Ω is σ ( A ) = ˆ ∞−∞ H n − ( A ∩ { ( X, t ) ∈ ∂ Ω } ) d t, (2.24) where H n − is the n − dimensional Hausdorff measure on the Lipschitz boundary ∂ Ω τ . We consider solvability of the L p Dirichlet boundary value problem with respectto this measure σ . The measure σ may not be comparable to the usual surfacemeasure on ∂ Ω: in the t -direction the functions φ j from definition 2.10 are only 1 / A ⊂ Z j , where Z j is an ℓ -cylinder, we have H n ( A ) ∼ σ ( { ( φ j ( x, t ) , x, t ) : ( x, t ) ∈ A } ) , (2.25)where the constants in (2.25), by which these measures are comparable, only dependon ℓ of the character ( ℓ, η, N, d ) of the domain Ω. If Ω has a smoother boundary,such as Lipschitz (in all variables) or better, then the measure σ is comparableto the usual n -dimensional Hausdorff measure H n . In particular, this holds for aparabolic cylinder Ω = O × R . Corollary 2.14. Let Ω be defined as in definition 2.10 by a family of functions { φ j } , φ j : Q d → R . Then there exists an extended family { Φ j } , Φ j : R n − × R → R ,such that(i) n Φ j | Q r o still describes Ω , as in definition 2.10, but with character ( ℓ, η, ˜ N , r ) instead of ( ℓ, η, N, d ) , where ˜ N ≥ N and r ≤ r ≤ d is from Theorem 2.8;(ii) k∇ Φ j k ∗ . ε η − ε + ηℓ , and(iii) k D Φ j k ∗ . ε η − ε + ηℓ .Proof. This follows from Theorem 2.8 and by tiling the support of each φ j intoparabolic cubes of size 8 r with enough overlap. (cid:3) Corollary 2.15. If Ω is a VMO type domain then we may take η arbitrarily smallin Corollary 2.14, or in (2.22) and (2.23) of definition 2.10, by reducing r . Pullback Transformation and Carleson Condition. We now briefly re-call the pullback mapping of Dahlberg-Kenig-Nečas-Stein on the upper half-space U ρ : U → Ω (c.f. [HL96, HL01]) in the setting of parabolic equations defined by ρ ( x , x, t ) = ( x + P γx φ ( x, t ) , x, t ) . (2.26)For simplicity assumeΩ = { ( x , x, t ) ∈ R × R n − × R : x > φ ( x, t ) } (2.27)where φ ( x, t ) : R n − × R → R and satisfies conditions (3) and (4) of definition 2.10.This transformation maps the upper half-space U = { ( x , x, t ) : x > , x ∈ R n − , t ∈ R } (2.28)into Ω and allows us to consider the L p solvability of the PDE (1.1) in the upperhalf-space instead of in the original domain Ω.To complete the definition of the mapping ρ we define a parabolic approximationto the identity P to be an even non-negative function P ( x, t ) ∈ C ∞ ( Q (0 , x, t ) ∈ R n − × R , with ´ P ( x, t ) d x d t = 1 and set P λ ( x, t ) := λ − ( n +1) P (cid:18) xλ , tλ (cid:19) . Let P λ φ be the convolution operator P λ φ ( x, t ) := ˆ R n − × R P λ ( x − y, t − s ) φ ( y, s ) d y d s then P satisfies for constants γ lim ( y ,y,s ) → (0 ,x,t ) P γy φ ( y, s ) = φ ( x, t )and ρ defined in (2.26) extends continuously to ρ : U → Ω. The usual surfacemeasure on ∂U is comparable with the measure σ defined by (2.24) on ∂ Ω.Suppose that v = u ◦ ρ and f v = f ◦ ρ then (1.1) transforms to a new PDE forthe variable v ( v t = div( A v ∇ v ) + B v · ∇ v in U,v = f v on ∂U, (2.29)where A v = [ a vij ( X, t )], B v = [ b vi ( X, t )] are ( n × n ) and (1 × n ) matrices.The precise relations between the original coefficients A and B and the new coef-ficients A v and B v are detailed in [Riv14, p. 448]. We note that if the constant γ > a vij , b vi : U → R are Lebesgue measur-able and A v satisfies the standard uniform ellipticity condition with constants λ v and Λ v , since the original matrix A did. Definition 2.16. Let Ω be a parabolic domain from definition 2.10. For ( Y, s ) ∈ ∂ Ω , ( X, t ) , ( Z, τ ) ∈ Ω and r > we write: B r ( X, t ) = { ( Z, τ ) ∈ R n × R : dist[( X, t ) , ( Z, τ )] < r } ,Q r ( X, t ) = { ( Z, τ ) ∈ R n × R : | x i − z i | < r for all ≤ i ≤ n − , | t − τ | / < r } , ∆ r ( Y, s ) = ∂ Ω ∩ B r ( Y, s ) , T (∆ r ) = Ω ∩ B r ( Y, s ) ,δ ( X, t ) = inf ( Y,s ) ∈ ∂ Ω dist[( X, t ) , ( Y, s )] . Definition 2.17 (Carleson measure) . A measure µ : Ω → R + is a Carleson meas-ure if there exists a constant C = C ( d ) such that for all r ≤ d and all surface balls ∆ r µ ( T (∆ r )) ≤ Cσ (∆ r ) . (2.30) ARABOLIC L p DIRICHLET PROBLEM AND VMO-TYPE DOMAINS 13 The best possible constant C is called the Carleson norm and is denoted by k µ k C,d .Occasionally, for brevity, we drop the d and just write k µ k C if the context is clear.We say that µ is a vanishing Carleson measure if k µ k C,d → as d → . When ∂ Ω is locally given as a graph of a function x = φ ( x, t ) in the coordinatesystem ( x , x, t ) and µ is a measure supported on { x > φ ( x, t ) } we can reformulatethe Carleson condition locally using the parabolic boundary cubes Q r and corres-ponding Carleson regions T ( Q r ). The Carleson condition (2.30) then becomes µ ( T ( Q r )) ≤ C | Q r | = Cr n +1 . (2.31)Note that the Carleson norms induced from (2.30) and (2.31) are not equal but arecomparable.We now return back to the pullback transformation and investigate the Carlesoncondition on the coefficients of A and B . The following result comes directly froma careful reading of the proofs of Lemma 2.8 and Theorem 7.4 in [HL96] combinedwith Theorems 2.1 and 2.3. Lemma 2.18. Let σ and θ be non-negative integers, α = ( α , . . . , α n − ) a multi-index with l = σ + | α | + θ , d a scale and fix γ . If φ : R n − × R → R satisfies for all x, y ∈ R n − , t, s ∈ R and for some positive constants ℓ and η | φ ( x, t ) − φ ( y, s ) | ≤ ℓ (cid:16) | x − y | + | t − s | / (cid:17) , k D φ k ∗ ≤ η (2.32) then the measure ν defined at ( x , x, t ) by d ν = (cid:18) ∂ l P γx φ∂x σ ∂x α ∂t θ (cid:19) x l +2 θ − d x d t d x is a Carleson measure on cubes of diameter ≤ d/ whenever either σ + θ ≥ or | α | ≥ , with ν [(0 , r ) × Q r ( x, t )] . η | Q r ( x, t ) | , where r ≤ d/ . Moreover, if l ≥ then at ( x , x, t ) , with x ≤ d/ , (cid:12)(cid:12)(cid:12)(cid:12) ∂ l P γx φ∂x σ ∂x α ∂t θ (cid:12)(cid:12)(cid:12)(cid:12) . η (1 + ℓ ) x − l − θ , (2.33) where the implicit constants depend on d, l, n . The drift term B v from the pullback transformation in (2.29) includes the term ∂∂t P γx φu x . From Lemma 2.18 with σ = | α | = 0, θ = 1, we see that x (cid:20) ∂∂t P γx φ ( x, t ) (cid:21) d X d t is a Carleson measure in U . Thus it is natural to expect thatd µ ( X, t ) = x | B v | ( X, t ) d X d t (2.34)is a Carleson measure in U and B v satisfies x | B v | ( X, t ) ≤ Λ B < k µ k / C . (2.35)Indeed, this is the case provided the original vector B satisfies the assumption thatd µ ( X, t ) = δ ( X, t ) " sup B δ ( X,t ) ( X,t ) | B | d X d t (2.36) is a Carleson measure in Ω. Here k µ k C depends on η and the Carleson normof (2.36).Similarly, for the matrix A v if we apply Lemma 2.18 and use the calculationsin [Riv14, §6] thend µ ( X, t )="(" x |∇ A v | + x | A vt | )( X, t ) d X d t (2.37)is a Carleson measure in U and A v satisfies( x |∇ A v | + x | A vt | )( X, t ) ≤ k µ k C (2.38)for almost everywhere ( X, t ) ∈ U provided the original matrix A satisfies thatd µ ( X, t )="" δ ( X, t ) " sup B δ ( X,t ) ( X,t ) |∇ A | + δ ( X, t ) " sup B δ ( X,t ) ( X,t ) | ∂ t A | d X d t (2.39)is a Carleson measure in Ω.We note that if both k µ k C,r and η are small then so too are the Carleson norms k µ k C,r and k µ k C,r of the matrix A v and vector B v , at least if we restrict ourselvesto small Carleson regions r ≤ d ; this comes from Theorem 2.8 and Corollaries 2.14and 2.15. Then by Lemma 2.18 we see that k µ k C,r and k µ k C,r only="if (!window.__cfRLUnblockHandlers) return false; " depend on η and k µ k C,r on Carleson regions of size r ≤ d . In particular they are small if both η and k µ k C,r are small. It further follows by Corollary 2.15 that we can make k µ k C,r and k µ k C,r as small as we like if µ is a vanishing Carleson norm and the domainΩ is of VMO type.Observe that condition (2.39) is slightly stronger than (1.6), which we claimed toassume in Theorem 1.1. We replace condition (2.39) by the weaker condition (1.6)later via perturbation results of [Swe98]. Definition 2.19.< p data-cf-modified-42563a8e35efde8a09487b78->
We define ρ j : U → Z j to be the local pullback mapping in Z j associated to the function Φ j in Theorem 2.8, the extension of φ j from defini-tion 2.10. Remark 2.20. By [BZ17] and its adaptation to the setting of admissible domainsin [DH16, §2.3] , one may construct a ‘proper generalised distance’ globally when η in the character of the domain is small. The smallness of η in the character of thedomain is used to guarantee that overlapping coordinate charts, generated by a localconstruction, are almost parallel. We may then use the result of [BZ17, Theorem5.1] to show there exists a domain Ω ε of class C ∞ , a homeomorphism f ε : Ω → Ω ε such that f ε ( ∂ Ω) = ∂ Ω ε and f ε : Ω → Ω ε is a C ∞ diffeomorphism. Parabolic Non-tangential Cones, Maximal Functions and p -adaptedSquare and Area Functions. We proceed with the definition of parabolic non-tangential cones and define the cones in a (local) coordinate system where Ω = { ( x , x, t ) : x > φ ( x, t ) } , which also applies to the upper half-space U . Definition 2.21. For a constant a > , we define the parabolic non-tangential coneat a point ( x , x, t ) ∈ ∂ Ω as follows Γ a ( x , x, t ) = n ( y , y, s ) ∈ Ω : | y − x | + | s − t | / < a ( y − x ) , x < y o . We occasionally truncate the cone Γ at the height r Γ ra ( x , x, t ) = n ( y , y, s ) ∈ Ω : | y − x | + | s − t | / < a ( y − x ) , x < y < x + r o . ARABOLIC L p DIRICHLET PROBLEM AND VMO-TYPE DOMAINS 15 Definition 2.22 (Non-tangential maximal function) . For a function u : Ω → R ,the non-tangential maximal function N a ( u ) : ∂ Ω → R and its truncated version ata height r are defined as N a ( u )( x , x, t ) = sup ( y ,y,s ) ∈ Γ a ( x ,x,t ) | u ( y , y, s ) | ,N ra ( u )( x , x, t ) = sup ( y ,y,s ) ∈ Γ ra ( x ,x,t ) | u ( y , y, s ) | for ( x , x, t ) ∈ ∂ Ω . (2.40)The following p -adapted square function was introduced in [DPP07] and hasbeen modified appropriately for the parabolic setting. It is used to control thespatial derivatives of the solution. When p = 2 it is equivalent to the usual squarefunction and when p < |∇ u | | u | p − iszero whenever ∇ u vanishes. Definition 2.23 ( p -adapted square function) . For a function u : Ω → R , the p -adapted square function S p,a ( u ) : ∂ Ω → R and its truncated version at a height r are defined as S p,a ( u )( Y, s ) = ˆ Γ a ( Y,s ) |∇ u ( X, t ) | | u ( X, t ) | p − δ ( X, t ) − n d X d t ! /p ,S rp,a ( u )( Y, s ) = ˆ Γ ra ( Y,s ) |∇ u ( X, t ) | | u ( X, t ) | p − δ ( X, t ) − n d X d t ! /p . (2.41) By applying Fubini we also have k S p,a ( u ) k pL p ( ∂U ) ∼ ˆ U |∇ u | | u | p − x d x d x d t. (2.42) It is not know a priori if these integrals are locally integrable even for p > .However, Theorem 4.1 shows that these expressions makes sense and are finite forsolutions to (1.1) . We also need a p -adapted version of an object called the area function whichwas introduced in [DH16] and is used to control the solution in the time variable.Again when p = 2 this is just the area function of [DH16]. Definition 2.24 ( p -adapted area function) . For a function u : Ω → R , the p -adapted area function A p,a ( u ) : ∂ Ω → R and its truncated version at a height r aredefined as A p,a ( u )( Y, s ) = ˆ Γ a ( Y,s ) | u t | | u ( X, t ) | p − δ ( X, t ) − n d X d t ! /p ,A rp,a ( u )( Y, s ) = ˆ Γ ra ( Y,s ) | u t | | u ( X, t ) | p − δ ( X, t ) − n d X d t ! /p . (2.43) Also by Fubini k A p,a ( u ) k pL p ( ∂U ) ∼ ˆ U | u t | | u | p − x d x d x d t. (2.44) As before, it is not known a priori if these expressions are finite for solutions to (1.1) but in Lemma 4.5 we establish control of A p,a by S p, a and use the finiteness of S p,a from Theorem 4.1. L p Dirichlet Boundary Value Problem. We are now in the position todefine the L p Dirichlet boundary value problem. Definition 2.25 ([Aro68]) . We say that u is a weak solution to a parabolic operatorof the form (1.1) in Ω if u, ∇ u ∈ L (Ω) , sup t k u ( · , t ) k L (Ω t ) < ∞ and ˆ Ω ( − uφ t + A ∇ u · ∇ φ − φB · ∇ u ) d X d t = 0 for all φ ∈ C ∞ (Ω) . Definition 2.26. We say that the L p Dirichlet problem with boundary data in L p ( ∂ Ω , d σ ) is solvable if the unique solution u to (1.1) for any continuous boundarydata f decaying to as t → ±∞ satisfies the following non-tangential maximumfunction estimate k N ( u ) k L p ( ∂ Ω , d σ ) . k f k L p ( ∂ Ω , d σ ) , (2.45) with the implied constant depending only on the operator, n , p and Ω . Basic Results and Interior Estimates We now recall some foundational estimates that will be used. The followingresult is from [DH16], which was adapted from the elliptic result in [Din02]. Lemma 3.1. Let r > and < a < b . Consider the non-tangential maximalfunctions defined using two set of cones cones Γ ra and Γ rb . Then for any p > thereexists a constant C p > such that for all u : U → R N ra ( u ) ≤ N rb ( u ) and k N rb ( u ) k L p ( ∂U ) ≤ C p k N ra ( u ) k L p ( ∂U ) . Lemma 3.2 (A Cacciopoli inequality, see [Aro68]) . Let A and B satisfy (1.2) and (2.35) and suppose that u is a weak solution of (1.1) in Q r ( X, t ) with < r <δ ( X, t ) / . Then there exists a constant C = C ( λ, Λ , n ) such that r n sup Q r/ ( X,t ) u ! ≤ C sup t − r ≤ s ≤ t + r ˆ Q r ( X,t ) ∩{ t = s } u ( Y, s ) d Y + C ˆ Q r ( X,t ) |∇ u | d Y d s ≤ C r ˆ Q r ( X,t ) u ( Y, s ) d Y d s. Lemmas 3.4 and 3.5 in [HL01] give the following estimates for weak solutionsof (1.1). Lemma 3.3 (Interior Hölder continuity) . Let A and B satisfy (1.2) and (2.35) and suppose that u is a weak solution of (1.1) in Q r ( X, t ) with < r < δ ( X, t ) / .Then for any ( Y, s ) , ( Z, τ ) ∈ Q r ( X, t ) | u ( Y, s ) − u ( Z, τ ) | ≤ C (cid:18) | Y − Z | + | s − τ | / r (cid:19) α sup Q r ( X,t ) | u | , where C = C ( λ, Λ , n ) , α = α ( λ, Λ , n ) , and < α < . Lemma 3.4 (Harnack inequality) . Let A and B satisfy (1.2) and (2.35) and sup-pose that u is a weak non-negative solution of (1.1) in Q r ( X, t ) , with < r <δ ( X, t ) / . Suppose that ( Y, s ) , ( Z, τ ) ∈ Q r ( X, t ) then there exists C = C ( λ, Λ , n ) such that, for τ < s , u ( Z, τ ) ≤ u ( Y, s ) exp (cid:20) C (cid:18) | Y − Z | | s − τ | + 1 (cid:19)(cid:21) . We state a version of the maximum principle from [DH16] that is a modificationof [HL01, Lemma 3.38]. ARABOLIC L p DIRICHLET PROBLEM AND VMO-TYPE DOMAINS 17 Lemma 3.5 (Maximum Principle) . Let A and B satisfy (1.2) and (2.35) , and let u and v be bounded continuous weak solutions to (1.1) in Ω . If | u | , | v | → uniformlyas t → −∞ and lim sup ( Y,s ) → ( X,t ) ( u − v )( Y, s ) ≤ for all ( X, t ) ∈ ∂ Ω , then u ≤ v in Ω . Remark 3.6 ([DH16]) . The proof of Lemma 3.38 from [HL01] works given theassumption that | u | , | v | → uniformly as t → −∞ . Even with this additionalassumption, the lemma as stated is sufficient for our purposes. We shall mostly useit when u ≤ v on the boundary of Ω ∩ { t ≥ τ } for a given time τ . Obviously then theassumption that | u | , | v | → uniformly as t → −∞ is not necessary. Another casewhen the lemma as stated here applies is when u | ∂ Ω , v | ∂ Ω ∈ C ( ∂ Ω) , where C ( ∂ Ω) denotes the class of continuous functions decaying to zero as t → ±∞ . This classis dense in any L p ( ∂ Ω , dσ ) , < p < ∞ allowing us to consider an extension of thesolution operator from C ( ∂ Ω) to L p . Improved Regularity for p -adapted square function Here we extend recent work of [DP16] for complex coefficient elliptic equations tothe real parabolic setting. The goal is to obtain a improved regularity result for weaksolutions of (1.1) implying that |∇ u | | u | p − belongs to L loc (Ω) when 1 < p < p -adapted square function S p,a is well defined atalmost every boundary point. Theorem 4.1 (c.f. [DP16, Theorem 1.1]) . Suppose u ∈ W , (Ω) is a weak solutionto L u = u t , where L u = div( A ∇ u ) + B ∇ u , A is bounded and elliptic, B is locallybounded and satisfies δ ( X, t ) | B ( X, t ) | ≤ K (4.1) for some uniform constant K > . Then for any parabolic ball B r ( X, t ) ⊂ Ω and p, q ∈ (1 , ∞ ) we have the following improvement in regularity B r ( X,t ) | u | p ! /p ≤ C ε B r ( X,t ) | u | q ! /q + ε B r ( X,t ) | u | ! / . (4.2) Here the constant C ε only depends on p , q , ε , n , λ , Λ , and K but not on u , ( X, t ) or r . In addition, for all < p < ∞ r B r ( X,t ) |∇ u | | u | p − ≤ C ε B r ( X,t ) | u | p + ε B r ( X,t ) | u | ! p/ , (4.3) where again the constant only depends on ε , p , n , the ellipticity constants of A , and K . This also shows that | u | ( p − / ∇ u ∈ L (Ω) . Remark 4.2. If q ≥ in (4.2) or if p ≥ in (4.3) then one can take ε = 0 becausethe L averages can be controlled by the first term on the right hand side of theseinequalities. We focus only on the case 1 < p < p ≥ < p < Lemma 4.3 (c.f. [DP16, Lemma 2.7]) . Let u be a weak solution to L u = u t in Ω for A elliptic and bounded, and B bounded satisfying (4.1) . Then for any p < ,any ball B r ( X, t ) with r < δ ( X, t ) / , and any ε > r ˆ B r ( X,t ) |∇ u | | u | p − ≤ C ε B r ( X,t ) | u | p + ε B r ( X,t ) | u | ! p/ (4.4) and B r ( X,t ) | u | ! / ≤ C ε B r ( X,t ) | u | p ! /p + ε B r ( X,t ) | u | ! / , (4.5) where the constants only depend on n , ε , λ , Λ and K . In particular, | u | ( p − / ∇ u ∈ L (Ω) .Proof. We start by assuming that A and B are smooth then the solution u to L u = u t is smooth. We prove the above inequalities with constants that do notdepend on the smoothness of A or B and then remove the smoothness assumptionat the end of the proof via the method of [HL01]. To simplify notation we suppressthe argument of the ball B r ( X, t ).Let ρ δ ( s ) = ( δ ( p − / ≤ s ≤ δs ( p − / s > δ. (4.6)The choice of cut off function ρ δ in this proof is inspired by [Lan99, p. 311], [CM05,p. 1088]. We multiply L u = u t by ρ δ ( | u | ) u and integrate by parts to obtain ˆ B r ∇ (cid:0) ρ δ ( | u | ) u (cid:1) A ∇ u = ˆ B r ρ δ ( | u | ) uu t + ˆ B r ρ δ ( | u | ) B · ∇ u + ˆ ∂B r ( ρ δ ( | u | )) ν · A ∇ u d σ ( y, s ) , (4.7)where ν is the outer unit normal to B r . Consider E δ = { u > δ } then the left handside of (4.7) is ˆ B r ∇ (cid:0) ρ δ ( | u | ) u (cid:1) A ∇ u = δ p − ˆ B r \ E δ ∇ u · A ∇ u + ˆ B r ∩ E δ A ∇ u · ∇ (cid:0) | u | p − u (cid:1) (4.8)and by ellipticity of A on the open set B r ∩ E δ we have for some λ ′ > λ ′ ˆ B r ∩ E δ | u | p − |∇ u | ≤ ˆ B r ∩ E δ A ∇ u · ∇ (cid:0) | u | p − u (cid:1) . (4.9)Our strategy is to let δ → B r \ E δ tend to 0.First, we use the following result from [Lan99]. They proved if u ∈ C (cid:0) B r (cid:1) and u = 0 on ∂B r then for q > − δ → δ q ˆ B r \ E δ |∇ u | = 0 . (4.10)To deal with the boundary integral in (4.7) we note that equations (4.7) to (4.9)remain valid for any enlarged ball B αr for 1 ≤ α ≤ / 4. We write (4.7) for every B αr and then average in α over the interval [1 , / B r/ \ B r . Therefore, λ ′ ˆ B r ∩ E δ | u | p − |∇ u | ≤ sup α ∈ [1 , / (cid:12)(cid:12)(cid:12)(cid:12) ˆ B αr ρ δ ( | u | ) uu t (cid:12)(cid:12)(cid:12)(cid:12) + sup α ∈ [1 , / (cid:12)(cid:12)(cid:12)(cid:12) ˆ B αr ρ δ ( | u | ) uB · ∇ u (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r − ˆ B αr/ \ B r ρ δ ( | u | ) uν · A ∇ u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + o (1)= I + II + III + o (1) , where o (1) contains the integral over B αr \ E δ , which tends to 0 as δ → 0. Webound II and III as [DP16] II + III ≤ C ε r − ˆ B r/ | u | p + εr p − ˆ B r/ |∇ u | p + o (1) . ARABOLIC L p DIRICHLET PROBLEM AND VMO-TYPE DOMAINS 19 Now we turn to I and use the same idea as the proof of (4.10) in [Lan99, (3.3)]to show I converges as expected. By splitting the integral with the set E δ , usingthe fact δ p − ≤ | u | p − on B αr \ E δ (since p < u , whichimplies | u | p − uu t ∈ L ( B αr ), we obtain ˆ B αr ρ δ ( | u | ) uu t = ˆ B αr ∩ E δ | u | p − uu t + δ p − ˆ B αr \ E δ uu t ≤ ˆ B αr ∩ E δ | u | p − uu t + ˆ B αr \ E δ | u | p − uu t ≤ ˆ B αr | u | p − | u t | < ∞ . Therefore by the dominated convergence theorem ˆ B αr ρ δ ( | u | ) uu t → ˆ B αr | u | p − uu t . (4.11)We change from working with balls to integrating over parabolic cubes Q αr anddenote by Q αr | s the cube Q αr restrict to the hypersurface { t = s } . Using thefundamental theorem of calculus we obtain in the limit that ˆ B αr | u | p − uu t ∼ ˆ B αr ∂∂t ( | u | p ) d t d X ≤ ˆ Q αr ∂∂t ( | u | p ) d t d X = ˆ t +( αr ) t − ( αr ) dd t ˆ Q αr | s | u | p d X d s ≤ k u k pL pX ( Q αr | t αr )2 ) + k u k pL pX ( Q αr | t − ( αr )2 ) . (4.12)Observe that (4.12) holds for all time restricted cubes Q αr | t ± ( αr ) with α ∈ [1 , . ˆ B αr | u | p − uu t . r ˆ Q . αr | u | p d X d t. Since Q . αr ⊂ B r , in the limit as δ → I . r ˆ B r | u | p d X d t. Therefore grouping the estimates together we have the following bound λ ′ ˆ B r ∩ E δ | u | p − |∇ u | . C ε r − ˆ B r | u | p + εr p − ˆ B r/ |∇ u | p + o (1) . (4.13)We let δ → A and B .Finally, since no constants depend on the smoothness of A or B , we can removethe smoothness assumption by the same argument as in [HL01]. We suppose A is just elliptic and bounded, and B satisfies (4.1) then we approximate A and B by smooth matrices and vectors respectively. For each smooth approximation wehave (4.4) and (4.5) and then passing to the limit we obtain analogous estimatesfor W , solutions u of L u = u t with the constants having the same dependence asbefore. (cid:3) It follows that the p -adapted square function S p,a is well defined. [DH16] alsoconsidered an area function and established [DH16, Lemma 5.2] that this areafunction can be controlled by the usual square function. The case 1 < p < u . We fix a boundary point ( Y, s ) ∈ ∂ Ω and consider A p,a ( Y, s ). Clearly, the non-tangential cone Γ a ( Y, s ) can be covered by non-overlapping collection of Whitneycubes { Q i } with the following properties:Γ a ( Y, s ) ⊂ [ i Q i ⊂ Γ a ( Y, s ) , r i := diam( Q i ) ∼ dist( Q i , ∂ Ω) , Q i ⊂ Ω , (4.14)and the cubes { Q i } having only finite overlap. It follows that[ A p,a ( Y, s )] p . X i ( r i ) − n ˆ Q i | u t | u p − d X d t (4.15) . X i ( r i ) − n ˆ Q i |∇ u | u p − + (cid:0) |∇ A | + | B | (cid:1) |∇ u | u p − d X d t. We need the following estimate on each Q i . Lemma 4.4. Assume the ellipticity condition (1.2) and that the coefficients A and B of (1.1) satisfy the conditions |∇ A ( X, t ) | ≤ K/δ ( X, t ) and | B ( X, t ) | ≤ K/δ ( X, t ) , for some uniform constant K > . Then for all non-negative solutions u of (1.1) and any parabolic cube Q such that Q ⊂ Ω we have the following estimate ˆ Q |∇ u | u p − d X d t . r − ˆ Q |∇ u | u p − d X d t, (4.16) where r = diam( Q ) .Proof. Since we assume differentiability of the matrix A in the spatial variables wemay also assume that A is symmetric. Let us denote by W = ( w k ), where w k = ∂ k u for k = 0 , , . . . , n − 1. Differentiating (1.1) we obtain the following PDE for each w k ( w k ) t − div( A ∇ w k ) = div(( ∂ k A ) W ) + ∂ k ( B · W ) . (4.17)We multiply (4.17) by w k u p − ζ , integrate over 2 Q and integrate by parts. Here0 ≤ ζ ≤ Q , vanishing outside 2 Q andsatisfying r |∇ ζ | + r | ζ t | ≤ C for some C > Q . This gives ˆ Q ( w k ) t w k u p − ζ d X d t + ˆ Q a ij ( ∂ j w k ) ∂ i ( w k u p − ζ ) d X d t = − ˆ Q ( ∂ k a ij ) w j ∂ i ( w k u p − ζ ) d X d t − ˆ Q b i w i ∂ k ( w k u p − ζ ) d X d t. (4.18) ARABOLIC L p DIRICHLET PROBLEM AND VMO-TYPE DOMAINS 21 We rearrange and group similar terms together12 ˆ Q h ( w k u p/ − ζ ) i t d X d t − p − ˆ Q w k u p − u t ζ d X d t + ˆ Q A (cid:16) ∇ ( w k ζ ) u p/ − (cid:17) · (cid:16) ∇ ( w k ζ ) u p/ − (cid:17) d X d t + ( p − ˆ Q A (cid:16) ∇ ( w k ζ ) u p/ − (cid:17) · (cid:16) ( ∇ u ) w k u p/ − ζ (cid:17) d X d t = ˆ Q | w k | u p − ζζ t d X d t + ˆ Q | w k | u p − A ∇ ζ · ∇ ζ d X d t − ˆ Q b i w i ∂ k ( w k ζ ) u p − ζ d X d t − ( p − ˆ Q b i w i (cid:16) ( ∂ k u ) w k u p/ − ζ (cid:17) u p/ − ζ d X d t − ˆ Q b i w i w k u p − ζζ k d X d t − ˆ Q ( ∂ k a ij ) w j w k u p − ζζ i d X d t − ˆ Q ( ∂ k a ij ) w j ( ∂ i w k ζ ) u p − ζ d X d t − ( p − ˆ Q ( ∂ k a ij ) w j (cid:16) ( ∂ i u ) w k u p/ − ζ (cid:17) u p/ − ζ d X d t. (4.19)All the terms after the equal sign are “error” terms since they either contain aderivative of ζ , or coefficients ∇ A or B . These will be handled using the Cauchy-Schwartz inequality and the estimates for |∇ A | , | B | ≤ K/r . The four main termsare on the left hand side of (4.19). The term that needs further work is the secondterm and we use the PDE (1.1) for u t . This gives − p − ˆ Q w k u p − u t ζ d X d t = − p − ˆ Q w k u p − div( A ∇ u ) ζ d X d t − p − ˆ Q w k u p − B · W ζ d X d t. (4.20)Again the second term will be an “error” term. For the first term we observe theequality u p − div( A ∇ u ) = div( A ( ∇ u ) u p − ) − ( p − A (( ∇ u ) u p/ − ) · (( ∇ u ) u p/ − ) . It follows (by integrating by parts) − p − ˆ Q w k u p − div( A ∇ u ) ζ d X d t = ( p − ˆ Q A ( ∇ ( w k ζ ) u p/ − ) · (( ∇ u ) w k u p/ − ζ ) d X d t + (2 − p )(3 − p )2 ˆ Q A (( ∇ u ) w k u p/ − ζ ) · (( ∇ u ) w k u p/ − ζ ) d X d t. (4.21) We now group all main terms together; these are the first, second and fourth termson the left-hand side of (4.19) and the terms of (4.21). This givesLHS of (4.19) = 12 ˆ Q (cid:20)(cid:16) w k u p/ − ζ (cid:17) (cid:21) t d X d t + ˆ Q A (cid:16) ∇ ( w k ζ ) u p/ − (cid:17) · (cid:16) ∇ ( w k ζ ) u p/ − (cid:17) d X d t + 2( p − ˆ Q A (cid:16) ∇ ( w k ζ ) u p/ − (cid:17) · (cid:16) ( ∇ u ) w k u p/ − ζ (cid:17) d X d t + (2 − p )(3 − p )2 ˆ Q A (cid:16) ( ∇ u ) w k u p/ − ζ (cid:17) · (cid:16) ( ∇ u ) w k u p/ − ζ (cid:17) d X d t = 12 ˆ Q (cid:20)(cid:16) w k u p/ − ζ (cid:17) (cid:21) t d X d t + (cid:18) − − p )3 − p (cid:19) ˆ Q A (cid:16) ∇ ( w k ζ ) u p/ − (cid:17) · (cid:16) ∇ ( w k ζ ) u p/ − (cid:17) d X d t + ˆ Q A (cid:18)q − p )3 − p (cid:2) ∇ ( w k ζ ) u p/ − (cid:3) − q (2 − p )(3 − p )2 (cid:2) ( ∇ u ) w k u p/ − ζ (cid:3)(cid:19) · (cid:18)q − p )3 − p (cid:2) ∇ ( w k ζ ) u p/ − (cid:3) − q (2 − p )(3 − p )2 (cid:2) ( ∇ u ) w k u p/ − ζ (cid:3)(cid:19) d X d t ≥ ˆ Q h ( w k u p/ − ζ ) i t d X d t + ( p − λ − p ˆ Q (cid:12)(cid:12)(cid:12) ∇ ( w k ζ ) u p/ − (cid:12)(cid:12)(cid:12) d X d t. (4.22)Here we have first completed the square (using symmetry of A ), and then usedthe ellipticity of the matrix A . The important point is that for all 1 < p < ( p − λ − p is positive.We also note that we could have completed the square differently and obtainedinstead of (4.22) the estimateLHS of (4.19) ≥ ˆ Q h ( w k u p/ − ζ ) i t d X d t + ( p − − p ) λ ˆ Q (cid:12)(cid:12)(cid:12) ( ∇ u ) w k u p/ − ζ (cid:12)(cid:12)(cid:12) d X d t. (4.23)It follows that we could average (4.22) and (4.23) and have both ˆ Q (cid:12)(cid:12)(cid:12) ∇ ( w k ζ ) u p/ − (cid:12)(cid:12)(cid:12) d X d t and ˆ Q (cid:12)(cid:12)(cid:12) ( ∇ u ) w k u p/ − ζ (cid:12)(cid:12)(cid:12) d X d t in the estimate with small positive constants.Now we briefly mention how all the error terms of (4.19), (4.20) and (4.22) canbe handled. Some can be immediately estimated from above by r − ˆ Q | W | u p − d X d t, where the scaling factor r − comes from the estimates on ∇ ζ , ζ t , |∇ A | and | B | . Forother terms (for example the first term of fifth line of (4.19) or the second term ofthe same line) we use Cauchy-Schwarz. One of the terms in the product will be (cid:18) r − ˆ Q | W | u p − d X d t (cid:19) / , while the other term is one of (cid:18) ˆ Q (cid:12)(cid:12)(cid:12) ∇ ( w k ζ ) u p/ − (cid:12)(cid:12)(cid:12) d X d t (cid:19) / or (cid:18) ˆ Q (cid:12)(cid:12)(cid:12) ( ∇ u ) w k u p/ − ζ (cid:12)(cid:12)(cid:12) d X d t (cid:19) / . ARABOLIC L p DIRICHLET PROBLEM AND VMO-TYPE DOMAINS 23 It follows using the ε -Cauchy-Schwarz inequality that we can hide these on theleft-hand side of (4.19). Finally, we put everything together by summing over all k and recalling that W = ∇ u . This gives for some constant ε = ε ( p, λ, n ) > ε → p → τ ˆ Q ∩{ t = τ } |∇ u | u p − d X + ε ˆ Q |∇ u | u p − d X d t + ε ˆ Q |∇ u | u p − d X d t ≤ Cr − ˆ Q |∇ u | u p − d X d t. (4.24)In particular (4.16) holds. (cid:3) After using (4.16) in (4.15) we can conclude the following. Lemma 4.5. Let u be a non-negative solution of (1.1) with matrix A satisfyingthe ellipticity hypothesis and the coefficients satisfying the bound |∇ A | , | B | ≤ K/δ .Then given a > there exists a constant C = (Λ , λ, a, K, p, n ) such that A p,a ( u )( X, t ) ≤ CS p, a ( u )( X, t ) . (4.25) From this we have the global estimate k A p,a ( u ) k pL p ( ∂ Ω) ≤ C k S p,a ( u ) k pL p ( ∂ Ω) . (4.26)As far as the proof goes, the calculations above clearly work for solutions u withuniform bound u ≥ ε > 0. Hence considering v ε = u + ε and then taking the limit ε → 0+ using Fatou’s lemma yields (4.25) for all non-negative u , where we haveused the convention that |∇ u | u p − = 0 whenever u = 0 and ∇ u = 0 with a similarconvention for the second gradient in A p,a .5. Bounding the p -adapted square function by the non-tangentialmaximum function We slightly abuse notation and only work on a Carleson region T (∆ r ) in theupper half space U even though we formulate the following lemmas on any ad-missible domain Ω. The equivalence of these formulations via the pullback map ρ is discussed in section 2.2 and [DH16], and hence we omit the details. We startwith a local bound of the p -adapted square function by the non-tangential maximalfunction. Lemma 5.1. Let Ω be an admissible domain from definition 2.10 with character ( ℓ, η, N, d ) . Let < p < and u be a non-negative solution of (1.1) , with theCarleson conditions (1.7) and (1.8) on the coefficients A and B . Then there existsa constant C = C ( λ, Λ , N, C ) such that for any solution u with boundary data f on any ball ∆ r ⊂ ∂ Ω with r ≤ min { d/ , d/ (4 C ) } we have ˆ T (∆ r ) |∇ u | | u | p − x d x d x d t ≤ C (1 + k µ k )(1 + ℓ ) ˆ ∆ r (cid:0) N r (cid:1) ( u ) d x d t. (5.1)In addition we have the following global result. Lemma 5.2. Let Ω be an admissible domain with smooth boundary ∂ Ω . Let we have C ˆ r / ˆ ∂ Ω |∇ u | | u | p − x d x d t d x + 2 r ˆ r ˆ ∂ Ω u p ( x , x, t ) d x d t d x ≤ ˆ ∂ Ω u p ( r , x, t ) d x d t + ˆ ∂ Ω u p (0 , x, t ) d x d t + C (cid:16) k µ k C, r + k µ k C, r + k µ k / C, r (cid:17) ˆ ∂ Ω (cid:0) N r ( u ) (cid:1) p d x d t. (5.2) Proof of Lemmas 5.1 and 5.2. Let Q r ( y, s ) be a parabolic cube on the boundarywith r < d and let ζ be a smooth cut off function independent of the x variable.As long as there is no ambiguity we suppress the argument of Q r and extensivelyuse the Einstein summation convention. Let ζ be supported in Q r , equal 1 in Q r and satisfy the estimate r |∇ ζ | + r | ζ t | ≤ C for some constant C and.We start by estimating ˆ r ˆ Q r | u | p − a ij a ( ∂ i u )( ∂ j u ) ζ x d x d t d x , (5.3)where by ellipticity we have λ Λ ˆ r ˆ Q r |∇ u | | u | p − x d x d t d x ≤ ˆ r ˆ Q r | u | p − a ij a ( ∂ i u )( ∂ j u ) ζ x d x d t d x . Now we integrate by parts whilst noting that ν = (1 , , , . . . , 0) since the domainis { x > } . ˆ r ˆ Q r | u | p − a ij a ( ∂ i u )( ∂ j u ) ζ x d x d t d x = 1 p ˆ Q r a j a ∂ j ( | u ( r, x, t ) | p ) rζ d x d t − ˆ r ˆ Q r a | u | p − u∂ i ( a ij ∂ j u ) ζ x d x d t d x − ˆ r ˆ Q r ∂ i (cid:18) a (cid:19) | u | p − ua ij ∂ j uζ x d x d t d x − ˆ r ˆ Q r a ij a | u | p − u ( ∂ j u ) ζ∂ i ζx d x d t d x − ˆ r ˆ Q r a j a | u | p − u ( ∂ j u ) ζ d x d t d x − ˆ r ˆ Q r a ij a ∂ i (cid:0) | u | p − (cid:1) u ( ∂ j u ) ζ d x d t d x = I + II + III + IV + V + V I (5.4)Our strategy is to further estimate all these terms and then group similar termstogether. First consider II , we use that u is a solution to (1.1) II = − ˆ r ˆ Q r a | u | p − uu t ζ x d x d t d x + ˆ r ˆ Q r a | u | p − ub i ∂ i uζ x d x d t d x = II + II . ARABOLIC L p DIRICHLET PROBLEM AND VMO-TYPE DOMAINS 25 Using the identity 2 x = ∂ x we integrate by parts in x to obtain II = − ˆ r ˆ Q r a | u | p − uu t ζ ∂ x d x d t d x = − ˆ Q r a | u ( r, x, t ) | p − u ( r, x, t ) u t ( r, x, t ) ζ r d x d t + 12 ˆ r ˆ Q r ∂ (cid:18) a (cid:19) | u | p − uu t ζ x d x d t d x + p − ˆ r ˆ Q r a | u | p − ∂ uu t ζ x d x d t d x + 12 ˆ r ˆ Q r a | u | p − u∂ ∂ t uζ x d x d t d x = II + II + II + II . Consider the boundary term II and we integrate by parts in tII = − ˆ Q r a | u ( r, x, t ) | p − ∂ t (cid:0) u ( r, x, t ) (cid:1) ζ r d x d t = 14 ˆ Q r ∂ t (cid:18) a (cid:19) | u ( r, x, t ) | p − u ( r, x, t ) ζ r d x d t + 12 ˆ Q r a | u ( r, x, t ) | p − u ( r, x, t ) ζζ t r d x d t + p − ˆ Q r a | u ( r, x, t ) | p − u ( r, x, t ) u t ( r, x, t ) ζ r d x d t = II + II + II . Since p < 2, so p − < 0, we can absorb II into II and save II to boundlater on.Considering II , we swap the order of differentiation on ∂ ∂ t u and integrate byparts in t to show II = 12 ˆ r ˆ Q r a | u | p − u∂ t ∂ uζ x d x d t d x = − ˆ r ˆ Q r ∂ t (cid:18) a (cid:19) | u | p − u∂ uζ x d x d t d x − p − ˆ r ˆ Q r a | u | p − u t ∂ uζ x d x d t d x − ˆ r ˆ Q r a | u | p − u∂ uζζ t x d x d t d x = II + II + II . Observe that II = − II so these terms cancel. We bound II by II = 12 ˆ r ˆ Q r ∂ t a a | u | p − u∂ uζ x d x d t d x . (cid:18) ˆ r ˆ Q r | A t | | u | p x ζ d x d t d x (cid:19) / (cid:18) ˆ r ˆ Q r |∇ u | | u | p − x ζ d x d t d x (cid:19) / . Two parts of II we have left to bound are II and II . Both of these integralsinvolve ζζ t and therefore if ζ is a partition of unity when we sum over that partitionthese terms sum to 0. The terms II and III are simply dealt with by II . (cid:18) ˆ r ˆ Q r | B | | u | p x ζ d x d t d x (cid:19) / (cid:18) ˆ r ˆ Q r |∇ u | | u | p − x ζ d x d t d x (cid:19) / and III . (cid:18) ˆ r ˆ Q r |∇ A | | u | p x ζ d x d t d x (cid:19) / (cid:18) ˆ r ˆ Q r |∇ u | | u | p − x ζ d x d t d x (cid:19) / . The integral in the term IV contains the terms ζ∂ i ζ and as before if ζ is apartition of unity then after summing this term cancels out. Therefore the termsthat we have yet to estimate are I , V and V I .We consider V in the two cases j = 0 and j = 0 separately. Since ζ is independentof x by the fundamental theorem of calculus V { j =0 } = − ˆ r ˆ Q r | u | p − u ( ∂ u ) ζ d x d t d x = − p ˆ r ˆ Q r ∂ (cid:0) | u | p ζ (cid:1) d x d t d x = 1 p ˆ Q r | u (0 , x, t ) | p ζ d x d t − p ˆ Q r | u ( r, x, t ) | p ζ d x d t. For the j = 0 case we use that ∂ x = 1 and integrate this case by parts in x V { j =0 } = − p ˆ r ˆ Q r a j a ∂ j ( | u | p ) ζ d x d t d x = − p ˆ r ˆ Q r a j a ∂ j ( | u | p ) ζ ∂ x d x d t d x = − p ˆ Q r a j a ∂ j ( | u ( r, x, t ) | p ) ζ r d x d t + 1 p ˆ r ˆ Q r a j a ∂ j ∂ ( | u | p ) ζ x d x d t d x + 1 p ˆ r ˆ Q r ∂ (cid:18) a j a (cid:19) ∂ j ( | u | p ) ζ x d x d t d x = V + V + V . The term V = − I { j =0 } so they cancel out. For V we integrate by parts in x j V = − X j =0 p ˆ Q r a j a ∂ ( | u ( r, x, t ) | p ) ζ r d x d t − p ˆ r ˆ Q r ∂ j (cid:18) a j a (cid:19) ∂ ( | u | p ) ζ x d x d t d x − p ˆ r ˆ Q r a j a ∂ ( | u | p ) ζ∂ j ζx d x d t d x = V + V + V .V and V are of the same type and can be estimated as III by (cid:12)(cid:12)(cid:12)(cid:12) ˆ r ˆ Q r ∇ (cid:18) a j a (cid:19) ∇ ( | u | p ) ζ x d x d t d x (cid:12)(cid:12)(cid:12)(cid:12) . ˆ r ˆ Q r | u | p − |∇ u ||∇ A | ζ x d x d t d x . (cid:18) ˆ r ˆ Q r |∇ A | | u | p ζ x d x d t d x (cid:19) / (cid:18) ˆ r ˆ Q r |∇ u | | u | p − ζ x d x d t d x (cid:19) / . The final term from (5.4) to estimate is V IV I = − ˆ r ˆ Q r a ij a ∂ i (cid:0) | u | p − (cid:1) u ( ∂ j u ) ζ d x d t d x = (2 − p ) ˆ r ˆ Q r a ij a | u | p − ( ∂ i u )( ∂ j u ) ζ d x d t d x L p DIRICHLET PROBLEM AND VMO-TYPE DOMAINS 27 and since 2 − p < V I in the left hand side of (5.4).We are now at the stage where we can group together all the similar terms andestimate them. There are 4 different types of terms: J = I { j =0 } + II + V { j =0 } + V J = II J = II + II + III + X j =0 V + X j =0 V J = II + II + IV + X j =0 V . We shall use the following standard result multiple times to deal with termscontaining |∇ A | , | A t | or | B | ; a reference for this is [Ste93, p. 59]. Let µ be aCarleson measure and U the upper half space then for any function u we have ˆ U | u | p d µ ≤ k µ k C k N ( u ) k pL p ( R n ) , (5.5)with a local version holding on Carleson boxes as well.First we consider J , which consists of boundary terms at (0 , x, t ) and ( r, x, t ). J = 1 p ˆ Q r ∂ ( | u ( r, x, t ) | p ) ζ r d x d t − ˆ Q r ∂ t a a | u ( r, x, t ) | p − u ( r, x, t ) ζ r d x d t + 1 p ˆ Q r | u (0 , x, t ) | p ζ d x d t − p ˆ Q r | u ( r, x, t ) | p ζ d x d t − X j =0 p ˆ Q r a j a ∂ ( | u ( r, x, t ) | p ) ζ r d x d t. The second term in J , originating from II , has the bound II = − ˆ Q r ∂ t a a | u ( r, x, t ) | p − u ( r, x, t ) ζ r d x d t ≤ λ ˆ Q r | A t || u ( r, x, t ) | p ζ r d x d t ≤ k µ k / C, r λ k N r ( u ) k pL p ( Q r ) .J = 12 ˆ r ˆ Q r ∂ (cid:18) a (cid:19) | u | p − uu t ζ x d x d t d x ≤ λ (cid:18) ˆ r ˆ Q r |∇ A | | u | p x ζ d x d t d x (cid:19) / (cid:18) ˆ r ˆ Q r | u t | | u | p − x ζ d x d t d x (cid:19) / ≤ λ (cid:16) k µ k C, r k N r ( u ) k pL p ( Q r ) (cid:17) / (cid:18) ˆ r ˆ Q r | u t | | u | p − x ζ d x d t d x (cid:19) / . With a constant C = C ( λ, Λ , n ) we can bound J by J ≤ C (cid:18) ˆ r ˆ Q r (cid:0) x |∇ A | + x | B | + x | A t | (cid:1) | u | p ζ d x d t d x (cid:19) / × (cid:18) ˆ r ˆ Q r |∇ u | | u | p − x ζ d x d t d x (cid:19) / ≤ C (cid:16) ( k µ k C, r + k µ k C, r ) k N r ( u ) k pL p ( Q r ) (cid:17) / (cid:18) ˆ r ˆ Q r |∇ u | | u | p − x ζ d x d t d x (cid:19) / . Finally, J consists of terms of the type ζ∂ t ζ or ζ∂ i ζ . Later we take ζ to be apartition of unity and so when we sum up over the partition all the terms in J sum to 0.Therefore after all these calculations ˆ r ˆ Q r | u | p − a ij a ( ∂ i u )( ∂ j u ) ζ x d x d t d x = J + J + J + J ≤ n Λ λ ˆ Q r ∂ ( | u ( r, x, t ) | p ) ζ r d x d t + ˆ Q r | u (0 , x, t ) | p ζ d x d t − ˆ Q r | u ( r, x, t ) | p ζ d x d t + k µ k / C, r λ k N r ( u ) k pL p ( Q r ) + 1 λ (cid:16) k µ k C, r k N r ( u ) k pL p ( Q r ) (cid:17) / (cid:18) ˆ r ˆ Q r | u t | | u | p − x ζ d x d t d x (cid:19) / + C (cid:16) ( k µ k C, r + k µ k C, r ) k N r ( u ) k pL p ( Q r ) (cid:17) / × (cid:18) ˆ r ˆ Q r |∇ u | | u | p − x ζ d x d t d x (cid:19) / + J . (5.6)By assuming that Ω is smooth as well as an admissible domain (definition 2.10)there exists a collar neighbourhood V of ∂ Ω in R n +1 such that Ω ∩ V can be globallyparametrised by (0 , r ) × ∂ Ω for some small r > 0, see remark 2.20 and [DH16]for details. Using definition 2.10, there is a collection of charts covering ∂ Ω withbounded overlap, say by M . We consider a partition of unity of these charts ζ j , with ζ j having the same definition, support and estimates as ζ before, and P j ζ j = 1everywhere. Therefore, when we sum (5.6) over this partition of unity the term onthe left hand side is bounded below by1Λ ˆ r ˆ ∂ Ω | u | p − ( A ∇ u · ∇ u ) x d x d t d x , which is comparable to the truncated p -adapted square function k S rp ( u ) k pL p ( ∂ Ω) .Therefore, remembering that after summing J = 0, for any ε > λ Λ k S rp ( u ) k pL p ( ∂ Ω) ∼ λ Λ ˆ r ˆ ∂ Ω | u | p − |∇ u | x d x d t d x ≤ n Λ λ ˆ ∂ Ω ∂ ( | u ( r, x, t ) | p ) r d x d t + ˆ ∂ Ω | u (0 , x, t ) | p d x d t − ˆ ∂ Ω | u ( r, x, t ) | p d x d t + M k µ k / C, r λ k N r ( u ) k pL p ( ∂ Ω) + k µ k C, r ελ k N r ( u ) k pL p ( ∂ Ω) + ε ˆ r ˆ ∂ Ω | u t | | u | p − x ζ d x d t d x + C k µ k C, r + k µ k C, r ε k N r ( u ) k pL p ( ∂ Ω) + ε ˆ r ˆ ∂ Ω |∇ u | | u | p − x ζ d x d t d x . (5.7)By applying Lemma 4.5 to the p -adapted area function in (5.7) we see that the p -adapted square function on the right hand side of (5.7) is always multiplied by ε .By choosing ε small enough we can absorb this p -adapted square function into the ARABOLIC L p DIRICHLET PROBLEM AND VMO-TYPE DOMAINS 29 left hand side yielding C k S rp ( u ) k pL p ( ∂ Ω) ≤ ˆ ∂ Ω ∂ ( | u ( r, x, t ) | p ) r d x d t + ˆ ∂ Ω | u (0 , x, t ) | p d x d t − ˆ ∂ Ω | u ( r, x, t ) | p d x d t + C (cid:16) k µ k C, r + k µ k C, r + k µ k / C, r (cid:17) k N r ( u ) k pL p ( ∂ Ω) . (5.8)We integrate (5.8) in the r variable, average over [0 , r ] and use the identity ( ∂ | u | p ) x = ∂ ( | u | p x ) − | u | p to give C ˆ r ˆ ∂ Ω (cid:18) x − x r (cid:19) |∇ u | | u | p − d x d t d x + 2 r ˆ r ˆ ∂ Ω | u ( x , x, t ) | p d x d t d x ≤ ˆ ∂ Ω | u ( r , x, t ) | p d x d t + ˆ ∂ Ω | u (0 , x, t ) | p d x d t + C (cid:16) k µ k C, r + k µ k C, r + k µ k / C, r (cid:17) k N r ( u ) k pL p ( ∂ Ω) . (5.9)Finally truncating the first integral on the left hand side to [0 , r / 2] gives C ˆ r / ˆ ∂ Ω |∇ u | | u | p − x d x d t d x + 2 r ˆ r ˆ ∂ Ω | u ( x , x, t ) | p d x d t d x ≤ ˆ ∂ Ω | u ( r , x, t ) | p d x d t + ˆ ∂ Ω | u (0 , x, t ) | p d x d t + C (cid:16) k µ k C, r + k µ k C, r + k µ k / C, r (cid:17) k N r ( u ) k pL p ( ∂ Ω) . (5.10)The local estimate for Lemma 5.1 is obtained (exactly as in [DH16]) if we donot sum over all the coordinate patches but instead use the estimates obtained fora single boundary cube Q r in (5.6). (cid:3) We just need to control the first integral on the right hand side of (5.2) to achieveour goal of controlling the p -adapted square function. Thankfully this has alreadybeen done for us in the proof of [DH16, Cor. 5.3] which we encapsulate below. Lemma 5.3. Let Ω be as in Lemma 5.2 and u be a non-negative solution to (1.1) .For a small r > depending on the geometry the domain Ω there exists a constant C such that for ε = k µ k C, r + k µ k C, r + k µ k / C, r ˆ ∂ Ω u ( r , x, t ) p d x d t ≤ r ˆ r ˆ ∂ Ω u ( x , x, t ) p d x d t d x + Cε k N r ( u ) k pL p ( ∂ Ω) . Combining Lemmas 5.2 and 5.3 gives us the desired result. Corollary 5.4. Let Ω be as in Lemma 5.2 and u be a non-negative solution to (1.1) .For a small r > depending on the geometry the domain Ω there exists constants C , C > such that for ε = k µ k C, r + k µ k C, r + k µ k / C, r k S r / p ( u ) k pL p ( ∂ Ω) ∼ ˆ r / ˆ ∂ Ω |∇ u | | u | p − x d x d t d x ≤ C ˆ ∂ Ω | u (0 , x, t ) | p d x d t + C ε k N r ( u ) k pL p ( ∂ Ω) . (5.11) Bounding the non-tangential maximum function by the p -adaptedsquare function Our goal in this section has been vastly simplified due to [Riv03] proving alocal good- λ inequality. We use this to bound the non-tangential maximum func-tion by the p -adapted square function. We first bound the non-tangential max-imum function by the usual L based square function S ( u ) but a simple argumentfrom [DPP07, (3.41)] shows that for 1 < p < ε > k S r ( u ) k L p ( ∂ Ω) ≤ C ε k S rp ( u ) k L p ( ∂ Ω) + ε k N r ( u ) k L p ( ∂ Ω) , (6.1)with a local version of this statement holding as well.The good– λ inequality from [Riv03, p. 508] is expressed in the following lemma. Lemma 6.1. Let v be a solution to (2.29) and v ( X, t ) = 0 for some point ( X, t ) ∈ Q r . Let E = { (0 , x, t ) ∈ Q r : S ,a ( v ) ≤ λ } and q > then |{ (0 , x, t ) ∈ Q r : N a ( v ) > λ }| . |{ (0 , x, t ) ∈ Q r : S ,a ( v ) > λ }| + 1 λ q ˆ E S ,a ( v ) q d x d t. (6.2)If p ≥ Lemma 6.2. Let v be a solution to (2.29) in U and the coefficients of (2.29) satisfythe Carleson estimates (2.34) , (2.35) , (2.37) and (2.38) on all parabolic balls of size ≤ r . Then there exists a constant C such that for any r ∈ (0 , r / ˆ Q r N a/ ( v ) p d x d t ≤ C (cid:18) ˆ Q r A ,a ( v ) p d x d t + ˆ Q r S ,a ( v ) p d x d t (cid:19) + r n +1 | v ( A ∆ r ) | p , (6.3) where A ∆ r is a corkscrew point of the boundary ball ∆ r . That is a point r laterin time than the centre of ∆ r and at a distance comparable to r from the boundaryand r from the centre of the ball ∆ r .Proof. We first assume that v ( X, t ) = 0 for some ( X, t ) ∈ Q r and then we havethe good- λ inequality (6.2). The passage from this good- λ inequality to a local L p estimate is standard in the spirit of [FS72]. We remove the assumption v ( X, t ) = 0for the cost of adding the r n +1 | v ( A ∆ r ) | p term in the same way as [Riv03, DH16]. (cid:3) From this local estimate we can obtain the following global L p estimate by thesame proof as the global L estimate from [DH16, Theorem 6.3]. Theorem 6.3. Let u be a solution to (1.1) and the coefficients of (1.1) satisfy theCarleson estimates (2.36) and (2.39) then k N r ( u ) k L p ( ∂ Ω) . k S r ( u ) k L p ( ∂ Ω) + k u k L p ( ∂ Ω) (6.4) and by (6.1) k N r ( u ) k L p ( ∂ Ω) . k S rp ( u ) k L p ( ∂ Ω) + k u k L p ( ∂ Ω) . (6.5)7. Proof of Theorem 1.1 We only consider the case 1 < p < p ≥ 2. First assume either stronger Carleson condition of (2.39), or (1.7) and (1.8)holds. Therefore the Carleson conditions on the pullback coefficients (2.34), (2.35),(2.37) and (2.38) hold.Without loss of generality, by remark 2.20, we may assume that our domain issmooth. Consider f + = max { , f } and f − = max { , − f } where f ∈ C ( ∂ Ω) and ARABOLIC L p DIRICHLET PROBLEM AND VMO-TYPE DOMAINS 31 denote the corresponding solutions with these boundary data u + and u − respect-ively. Hence we may apply the Corollary 5.4 separately to u + and u − . By themaximum principle these two solutions are non-negative. It follows that for anysuch non-negative u we have k S rp ( u ) k pL p ( ∂ Ω) ≤ C k f k pL p ( ∂ Ω) + C ( k µ k / C + k µ k C ) k N r ( u ) k pL p ( ∂ Ω) and Theorem 6.3 gives k N r ( u ) k pL p ( ∂ Ω) ≤ C k f k pL p ( ∂ Ω) + C k S rp ( u ) k pL p ( ∂ Ω) , here k µ k C is the Carleson norm of (1.7) on Carleson regions of size ≤ r . As notedearlier, if for example Ω is of VMO type then size of µ appearing in this estimate willonly depend on the Carleson norm of coefficients on Ω, provided we only considersmall Carleson regions. Hence we can choose r small enough (depending on thedomain Ω) such that the Carleson norm after the pullback is say only twice theoriginal Carleson norm of the coefficients over all balls of size ≤ r .Since we are assuming k µ k C is small, clearly we also have k µ k C ≤ C k µ k / C .By rearranging these two inequalities and combining estimates for u + and u − , weobtain, for 0 < r ≤ r / k N r ( u ) k pL p ( ∂ Ω) ≤ C k f k pL p ( ∂ Ω) + C k µ k / C k N r ( u ) k pL p ( ∂ Ω) . By a simple geometric argument in [DH16] involving cones of different apertures,Lemmas 3.1 and 3.4 show there exists a constant M such that k N r ( u ) k pL p ( ∂ Ω) ≤ M k N r ( u ) k pL p ( ∂ Ω) . (7.1)It follows that if CM k µ k / C < / k N r ( u ) k pL p ( ∂ Ω) ≤ C k f k pL p ( ∂ Ω) , which is the desired estimate (for truncated version of non-tangential maximumfunction). The result with the non-truncated version of the non-tangential max-imum function N ( u ) follows as our domain is bounded in space and hence (7.1) canbe iterated finitely many times until the non-tangential cones have sufficient heightto cover the whole domain.Finally, we comment on how the Carleson condition (2.39) can be relaxed to theweaker condition (1.6). The idea is the same as [DH16, Theorem 3.1]. As shownthere, if the operator L satisfies the weaker condition (1.6), then it is possible (viamollification of coefficients) to find another operator L which is a small perturb-ation of the operator L and L satisfies (2.39). The solvability of the L p Dirichletproblem in the range 1 < p < L follows by our previous arguments. How-ever, as L is a small perturbation of the operator L we have by the perturbationargument of [Swe98] L p solvability of L as well.Finally, for larger values of p we use the maximum principle and interpolationto obtain solvability results in the full range 1 < p < ∞ . (cid:3) Appendix — proofs of results from section 2 Proof of Theorem 2.3. We begin by proving the equivalence of conditions (3) and (6)using ideas from [Str80] and write F = D φ where F is a tempered distribution. Let ϕ k = χ ˜ Q (0 , − χ ˜ Q ( e k )2 MARTIN DINDOŠ, LUKE DYER, AND SUKJUNG HWANG then for 1 ≤ k ≤ n − c ϕ k ( ξ, τ ) = 2 sin ( ξ k / ξ k − e − iτ iτ n − Y j = k − e − iξ j iξ j , c ϕ n ( ξ, τ ) = 2 sin ( τ / τ n − Y j =1 − e − iξ j iξ j , (8.1)with c ϕ k ( ξ, τ ) ∼ ξ k for small ξ k and 1 ≤ k ≤ n − 1. We let c ψ u = e i ( ξ, · u − k ( ξ, τ ) k and denote by ψ uρ ( x, t ) the usual parabolic dilation by ρ , that is ψ uρ ( x, t ) = ρ − ( n +1) ψ u ( x/ρ, t/ρ ) . It is worth noting that ( ϕ k ∗ ψ u ) ρ = ϕ kρ ∗ ψ uρ . Therefore we may rewrite condi-tion (6.a), by remark 2.4, assup Q r n − X k =1 | Q r | ˆ Q r ˆ u ∈ S n − ˆ r (cid:0) ψ uρ ∗ ϕ kρ ∗ F (cid:1) d ρρ d u d x d t ∼ B (6 .a ) . (8.2)Similarly if we let c ψ un = e i (0 ,τ ) · u − k ( ξ, τ ) k (8.3)then we may rewrite condition (6.b) assup Q r | Q r | ˆ Q r ˆ u ∈ S n − ˆ r (cid:0) ψ un,ρ ∗ F (cid:1) d ρρ d u d x d t ∼ B (6 .b ) . (8.4)The functions ϕ k ∗ ψ u and ψ un all satisfy the following conditions for some ε i > ˆ ψ d x d t = 0 , | ψ ( x, t ) | . k ( x, t ) k − n − − ε for k ( x, t ) k ≥ a > , | b ψ ( ξ, τ ) | . k ( ξ, τ ) k ε for k ( ξ, τ ) k ≤ , | b ψ ( ξ, τ ) | . k ( ξ, τ ) k − ε for k ( ξ, τ ) k ≥ . (8.5)Therefore if D φ = F ∈ BMO( R n ) then B (6 .a ) . k D φ k ∗ and B (6 .b ) . k D φ k ∗ by [Str80, Theorem 2.1]; this shows condition (3) implies condition (6).For the converse we proceed via an analogue of the proof of [Str80, Theorem 2.6].Consider ˆ θ ( ξ, τ ) = k ( ξ, τ ) k ˆ ζ ( ξ, τ ) , where ζ ∈ C ∞ ( R ). Let H be the dense subclass of continuous H functions g suchthat g and all its derivatives decay rapidly, see [Ste70, p. 225]. Via an analogueof [FS72, Theorem 3], [Str80, Lemma 2.7] by assuming conditions (6.a) and (6.b) if g ∈ H ( R n ) then for each 1 ≤ k ≤ n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ S n − ˆ ∞ ¨ R n − × R ψ uρ ∗ ϕ kρ ∗ F ( x, t ) θ ρ ∗ g ( x, t ) d x d t d ρρ d u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . B / (6 .a ) k g k H , (8.6)and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ S n − ˆ ∞ ¨ R n − × R ψ un,ρ ∗ F ( x, t ) θ ρ ∗ g ( x, t ) d x d t d ρρ d u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . B / (6 .b ) k g k H . (8.7) ARABOLIC L p DIRICHLET PROBLEM AND VMO-TYPE DOMAINS 33 For 1 ≤ k ≤ n − m k ( ξ, τ ) = ˆ S n − ˆ ∞ ˆ ψ u ( − ρξ, − ρ τ ) ˆ ϕ k ( − ρξ, − ρ τ ) k ( ξ, τ ) k ζ ( ρ k ( ξ, τ ) k ) d ρ d u,m n ( ξ, τ ) = ˆ S n − ˆ ∞ ˆ ψ un ( − ρξ, − ρ τ ) k ( ξ, τ ) k ζ ( ρ k ( ξ, τ ) k ) d ρ d u. (8.8)All of these functions m i are homogeneous of degree zero, smooth away from theorigin and the associated Fourier multipliers M k , for 1 ≤ k ≤ n , are Caldorón-Zygmund operators that preserve the class H and are bounded on H .The non-degeneracy condition from [CT75] on the family of functions { m k } nk =1 holds — that is the property that P k | m k ( rξ, r τ ) | does not vanish identically in r for ( ξ, τ ) = (0 , u k,j ( ξ, τ ) and positive numbers r j such that for all ( ξ, τ ) =(0 , n X k =1 j X j =1 m k,r j ( ξ, τ ) u k,j ( ξ, τ ) = 1 , (8.9)where m k,r j are as m k but with r j ρ replacing ρ in the arguments of ˆ ψ u , ˆ ϕ k and ˆ ψ u in (8.8) (but not ζ ).Let M k,j and U k,j be the associated Fourier multiplier operators to their re-spective multipliers m k,r j and u k,j then P P M k,j U k,j g = g for all g ∈ H .By [FS72, Theorem 3], [Str80, Lemma 2.7] there exists h k,j ∈ BMO( R n ) suchthat k h k,j k ∗ . B (6 .a ) or B (6 .b ), and ( h k,j , g ) = ( F, M k,j g ) for all g ∈ H . If wereplace g by U j,k g ∈ H in the previous identity and sum over j and k we obtain( h, g ) = ( F, g ) for all g ∈ H where h = P k,j U ∗ k,j h k,j ; furthermore by the BMOcondition on h k,j , k h k ∗ . B (6 .a ) + B (6 .b ). The identity (8.9) does not need to holdat the origin therefore ˆ h − ˆ F may be supported at the origin and hence F = h + p where p is a polynomial. Due to the assumption φ ∈ Lip(1 , / 2) clearly F must bea tempered distribution. Hence as in [Str80] we may conclude F = h ∈ BMO( R n ).This implies equivalence of conditions (3) and (6).Similarly we may prove the equivalence of conditions (4) and (5) to condition (3).The changes needed are outlined below.We first look at condition (5) ⇐⇒ condition (3). In this instance we replace theconvolutions ϕ k ∗ ψ u by ˆ ψ u ( ξ, τ ) = e i ( ξ, · u − − e − i ( ξ, · u k ( ξ, τ ) k , which corresponds to condition (5.a), and we keep the convolution ψ un as it is in (8.3).The same proof then goes through to give that condition (5) holds if and only ifcondition (3) holds with equivalent norms, as in (2.13).Condition (4) ⇐⇒ condition (3). This case is stated in [Riv03, Proposition 3.2].Again the proof proceeds as above with one convolutionˆ ψ u ( ξ, τ ) = e i ( ξ,τ ) · u − − e − i ( ξ,τ ) · u k ( ξ, τ ) k . (cid:3) Proof of Theorem 2.8. Without loss of generality we only consider the case η < η ≥ k D Φ k ∗ . η + ℓ requires a muchsimpler argument.By (2.20) there exists f ∈ C δ such that k∇ φ − f k ∗ ,Q d ≤ η and a scale 0 < r = r ( δ ) ≤ d such that k f k ∗ ,Q d ,r ≤ η. Q r Q R x ∈ R n − t ∈ R Figure 1. The reflection and tiling of the cube Q r ⊂ Q R defined in (8.10).Let d ′ = η min( r , r ) / r ≤ d ′ and Q r ⊂ Q d . Find a naturalnumber k such that R = 2 k r and Rη/ < r ≤ Rη . By our choice of d ′ the cube Q R which is an enlargement of Q r by a factor 2 k +1 is still contained in the originalcube Q d .It follows that k∇ φ k ∗ ,Q R . η andsup Q s = J s × I s Q s ⊂ Q R | Q s | ˆ Q s ˆ I s | φ ( x, t ) − φ ( x, τ ) | | t − τ | d τ d t d x ≤ η . Without loss of generality we may now assume that the cube Q R is centredat the origin (0 , 0) and that φ (0 , 0) = 0, since the BMO norm is invariant undertranslation and ignores constants. We first define ˜ φ as an extension in time viareflection and tiling of the cube Q r :˜ φ ( x, t ) = ( φ ( x, t ) t ∈ [ − r , r ] + 4 kr ,φ ( x, r − t ) t ∈ [ r , r ] + 4 kr , k ∈ Z . (8.10)See figure 1 on page 34 for an illustration of this. Clearly ˜ φ coincides with φ on Q r .It follows that ˜ φ is a function ˜ φ : {| x | ∞ < R } × R → R and ( ∇ ˜ φ ) Q r = ( ∇ φ ) Q r .Consider a cut off function ρ such that ρ ( x ) = ( | x | ∞ < r | x | ∞ > R, and |∇ ρ | . /R . η/r . Finally defineΦ = ˜ φρ + (1 − ρ )( x · ( ∇ ˜ φ ) Q r ) . (8.11)Clearly Φ is well defined on R n − × R as ρ = 0 outside the support of ˜ φ . We claimthat Φ satisfies ( i )-( iv ) of Theorem 2.8 which we establish in a sequence of lemmas ARABOLIC L p DIRICHLET PROBLEM AND VMO-TYPE DOMAINS 35 below. Observe also that from our definition of Φ we have ∇ Φ = (cid:0) ∇ ˜ φ − ( ∇ ˜ φ ) Q r (cid:1) ρ + ∇ ρ ( ˜ φ − x · ( ∇ ˜ φ ) Q r ) + ( ∇ ˜ φ ) Q r . (8.12) (cid:3) We start with couple of lemmas that allows us to reduce our claim to the dyadiccase; this is to make the geometry easier to handle. Lemma 8.1 ([Jon80, Lemma 2.3], c.f. [Str80, Theorem 2.8]) . Let f be defined on R n and sup Q | Q | ˆ Q | f − f Q | ≤ c ( η ) , (8.13) where the supremum is taken over all dyadic cubes Q ⊂ R n . Further, assume that sup Q ,Q | f Q − f Q | ≤ c ( η ) , (8.14) where the supremum is taken over all dyadic cubes Q , Q of equal edge length witha touching edge. Then k f k ∗ . c ( η ) . Below l ( Q s ) = s denotes the radius of a parabolic cube. Lemma 8.2 ([Jon80, Lemma 2.1 and pp. 44-45]) . Let f ∈ BM O ( Q ) and Q ⊂ Q ⊂ Q then | f Q − f Q | . log (cid:18) l ( Q ) l ( Q ) (cid:19) k f k ∗ ,Q . (8.15) Furthermore, the same proof in [Jon80] gives the following slightly stronger result | Q | ˆ Q | f − f Q | . log (cid:18) l ( Q ) l ( Q ) (cid:19) k f k ∗ ,Q . (8.16) If Q , Q ⊂ Q and l ( Q ) ≤ l ( Q ) but they are not necessarily nested then | f Q − f Q | . (cid:18) log (cid:18) l ( Q ) l ( Q ) (cid:19) + log (cid:20) Q , Q ) l ( Q ) (cid:21)(cid:19) k f k ∗ ,Q . (8.17) If the cubes Q , Q and Q are dyadic then we may replace BMO by dyadic BMO . There is a typo at the top of [Jon80, p. 45]. It should read l ( Q k ) ≤ l ( Q j ) (itcurrently reads the converse). Claim 8.3. Let ˜ φ be defined as in (8.10) , k∇ φ k ∗ ,Q R . η , and let Q be dyadic with r ≤ l ( Q ) ≤ R then | Q | ˆ Q |∇ ˜ φ − ∇ ˜ φ Q r | . ε η − ε . (8.18) Proof of claim. Let N ∈ N be such that l ( Q ) = 2 N l ( Q r ). Let { Q i } be the 2 N ( n − dyadic cubes that are translations of Q r and partition Q ∩ {| t | ≤ r } . Then byLemma 8.21 | Q | ˆ Q |∇ ˜ φ − ∇ ˜ φ Q r | = X i N | Q i || Q | | Q i | ˆ Q i |∇ ˜ φ − ∇ ˜ φ Q r |≤ X i N | Q i || Q | (cid:18) | Q i | ˆ Q i |∇ φ − ∇ φ Q i | + |∇ φ Q i − ∇ φ Q r | (cid:19) . ( η + η log(2 + R/r )) . η + η log(1 + 1 /η ) . ε η − ε . (cid:3) Lemma 8.4 ([Ste76]) . Let g, h ∈ L then | Q | ˆ Q | gh − ( gh ) Q | ≤ | Q | ˆ Q | g ( h − h Q ) | + | h Q || Q | ˆ Q | g − g Q | . (8.19) Proof. This small reduction is from [Ste76, p. 582]. First observe gh − ( gh ) Q = g ( h − h Q ) + h Q ( g − g Q ) + g Q h Q − ( gh ) Q and | g Q h Q − ( gh ) Q | = (cid:12)(cid:12)(cid:12)(cid:12) | Q | ˆ Q gh Q − | Q | ˆ Q gh (cid:12)(cid:12)(cid:12)(cid:12) ≤ | Q | ˆ Q | g ( h − h Q ) | hence (cid:12)(cid:12)(cid:12)(cid:12) | Q | ˆ Q | gh − ( gh ) Q | − | h Q || Q | ˆ Q | g − g Q | (cid:12)(cid:12)(cid:12)(cid:12) ≤ | Q | ˆ Q | g ( h − h Q ) | . (8.20) (cid:3) We can now prove property (iii) of Theorem 2.8. Lemma 8.5. Let Φ : R n → R be defined as in (8.11) with k∇ φ k ∗ ,Q R . η then ∇ Φ ∈ BMO( R n ) and for all < ε < k∇ Φ k ∗ . ε η − ε + ηℓ. (8.21) Proof. Recall ∇ Φ = (cid:0) ∇ ˜ φ − ( ∇ ˜ φ ) Q r (cid:1) ρ + ∇ ρ ( ˜ φ − x · ( ∇ ˜ φ ) Q r )+( ∇ ˜ φ ) Q r ; we can ignorethe constant term as the BMO norm doesn’t see it. Let ψ = ∇ ˜ φ − ( ∇ ˜ φ ) Q r and θ = ˜ φ − x · ( ∇ ˜ φ ) Q r . We want to bound k ρψ k ∗ and k∇ ρθ k ∗ . We first tackle the term k ρψ k ∗ . Step 1: (8.14) holds: sup Q ,Q | ( ρψ ) Q − ( ρψ ) Q | ≤ c ( η ) for Q , Q dyadic cubesof equal side length and with a touching edge.Since ˜ φ is the extension in the time direction by reflection and tiling (c.f. (8.10)),and Q , Q and Q r are all dyadic cubes we may assume that if l ( Q ) ≤ r then Q , Q ⊂ {| t | < r } , and if l ( Q ) > r then {| t | < r } ⊂ Q .If Q , Q ⊂ Q R then | ( ρψ ) Q − ( ρψ ) Q | . k ρψ k ∗ , dyadic , Q R . Therefore, if weshow (8.13) for f = ρψ then by Lemmas 8.2 and 8.4 clearly | ( ρψ ) Q − ( ρψ ) Q | . k ρψ k ∗ , dyadic , Q R ≤ k ψ k ∗ , dyadic , Q R ≤ k∇ ˜ φ k ∗ , dyadic , Q R . η. Now look at the other cases: Q ⊂ Q R and Q ∩ Q R = ∅ , or Q R ⊂ Q and Q ∩ Q R = ∅ . In both cases we wish to control | ( ρψ ) Q | . Step 1.a: Case Q ⊂ Q R , Q ∩ Q R = ∅ and l ( Q ) . Rηℓ . Q is small here and touches the boundary of Q R . This means that k ρ k L ∞ ( Q ) . l ( Q ) R since ρ is 0 outside Q R . Therefore we just apply the trivial bound | ( ρψ ) Q | ≤ k ρ k L ∞ ( Q ) k ψ k L ∞ ( Q ) . l ( Q ) R ℓ . η. Step 1.b: Case Q ⊂ Q R , Q ∩ Q R = ∅ and Rηℓ . l ( Q ) ≤ R .Since Q ⊂ Q R we have Rηℓ . l ( Q ) ≤ R . Q is dyadic so there exists N ∈ Z such that l ( Q ) = 2 N l ( Q r ). Step 1.b.i: N ≤ l ( Q ) ≤ l ( Q r ) and so by the reflection and tiling in time, (8.10),we may assume Q ⊂ {| t | ≤ r } and by Lemma 8.2 | ( ρψ ) Q | ≤ | ψ | Q = 1 | Q | ˆ Q |∇ φ − ∇ φ Q r | ≤ | Q | ˆ Q |∇ φ − ∇ φ Q | + |∇ φ Q − ∇ φ Q r | . η + η log (1 + ℓ ) + η log(1 + 1 /η ) . ε η − ε + η log (1 + ℓ ) . Step 1.b.ii: N > | ( ρψ ) Q | ≤ | ψ | Q = 1 | Q | ˆ Q |∇ ˜ φ − ∇ ˜ φ Q r | . ε η − ε . ARABOLIC L p DIRICHLET PROBLEM AND VMO-TYPE DOMAINS 37 Step 1.c: Case Q R ⊂ Q , Q ∩ Q R = ∅ so l ( Q ) ≥ R .Let N satisfy l ( Q ) = 2 N l ( Q R ), the number of dyadic generations separating Q and Q R . Then Q overlaps Q R (and its dyadic translates in time) exactly 2 N times. Therefore by claim 8.3, | ( ρψ ) Q | ≤ | ψ | Q ≤ N | Q | ˆ Q R |∇ ˜ φ −∇ ˜ φ Q r | ≤ N N ( n +1) | Q R | ˆ Q R |∇ ˜ φ −∇ ˜ φ Q r | . ε η − ε . Hence, modulo the unproved statement k ρψ k ∗ , dyadic , Q R . η we have shown | ( ρψ ) Q − ( ρψ ) Q | . ε η − ε + η log(1 + ℓ ) . Step 2: (8.13) holds, that is: k ρψ k ∗ , dyadic . c ( η ).To apply Lemma 8.4 we need to control two termssup Q dyadic k ρ k L ∞ ( Q ) | Q | ˆ Q | ψ − ψ Q | and sup Q dyadic | ψ Q || Q | ˆ Q | ρ − ρ Q | . Step 2.a: Estimating sup Q dyadic k ρ k L ∞ ( Q ) | Q | ˆ Q | ψ − ψ Q | .In all the following cases we bound k ρ k L ∞ ( Q ) ≤ Step 2.a.i: Case l ( Q ) ≤ r .As before, by the reflection and tiling in time, we may assume Q ⊂ {| t | ≤ r } and so ∇ ˜ φ = ∇ φ on Q . Hence1 | Q | ˆ Q | ψ − ψ Q | = 1 | Q | ˆ Q |∇ ˜ φ − ( ∇ ˜ φ ) Q | = 1 | Q | ˆ Q |∇ φ − ( ∇ φ ) Q | . η. Step 2.a.ii: Case r < l ( Q ) ≤ R .Applying claim 8.3 gives1 | Q | ˆ Q | ψ − ψ Q | ≤ | ψ | Q . ε η − ε . Step 2.a.iii: Case 2 R < l ( Q ).From step 1.c it follows that1 | Q | ˆ Q | ψ − ψ Q | ≤ | ψ | Q . ε η − ε . Step 2.b: Estimating sup Q dyadic | ψ Q || Q | ˆ Q | ρ − ρ Q | .We have the following three cases to consider. Step 2.b.i: Case Q ⊂ Q R , l ( Q ) ≤ r and Q ⊂ {| t | ≤ r } .Because the cube Q might not be touching the boundary we can’t follow step 1.aand bound | Q | ´ Q | ρ − ρ Q | by k ρ k L ∞ ( Q ) , which here is likely be 1. However, we canuse the mean value theorem and get a better bound. By the intermediate valuetheorem there exists ( z, τ ) ∈ Q such that ρ ( z ) = ρ Q and using that ρ is independentof time and |∇ ρ | . /R we have | ρ ( x ) − ρ Q | = | ρ ( x ) − ρ ( z ) | ≤ |∇ ρ | l ( Q ) . l ( Q ) R ≤ l ( Q ) r . Then applying Lemma 8.2 gives | ψ Q || Q | ˆ Q | ρ − ρ Q | . l ( Q ) r (cid:12)(cid:12)(cid:12)(cid:12) | Q | ˆ Q ∇ ˜ φ − ∇ ˜ φ Q r (cid:12)(cid:12)(cid:12)(cid:12) ≤ l ( Q ) r | Q | ˆ Q |∇ φ − ∇ φ Q r | . l ( Q ) r log (cid:18) rl ( Q ) (cid:19) η . η. Step 2.b.ii: Case Q ⊂ Q R and r < l ( Q ) ≤ R .This case is a straightforward application of claim 8.3 | ψ Q || Q | ˆ Q | ρ − ρ Q | ≤ | ψ Q | . ε η − ε . Step 2.b.iii: Case Q R ⊂ Q so l ( Q ) > R .This follows similarly to step 1.c; let N be defined as there and | ψ Q || Q | ˆ Q | ρ − ρ Q | ≤ | Q | (cid:12)(cid:12)(cid:12)(cid:12) ˆ Q ∇ φ − ∇ φ Q R (cid:12)(cid:12)(cid:12)(cid:12) ≤ N N ( n +1) k∇ φ k ∗ ,Q R ≤ η. Therefore by Lemma 8.1, k ρψ k ∗ . ε η − ε + η log(1 + ℓ ).It remains to tackle the harder piece ∇ ρθ = ∇ ρ ( ˜ φ − x · ∇ ˜ φ Q r ). Recall thatsupp( ∇ ρ ) = { r ≤ | x | ∞ ≤ R } . Step 3: (8.14) holds; that is: sup Q ,Q | ( ∇ ρθ ) Q − ( ∇ ρθ ) Q | ≤ c ( η ) where Q , Q are dyadic with a touching edge and l ( Q ) = l ( Q ).There are two different cases to consider:(1) Q ∩ supp( ∇ ρ ) = ∅ and Q ∩ supp( ∇ ρ ) = ∅ (2) Q ∩ supp( ∇ ρ ) = ∅ and Q ∩ supp( ∇ ρ ) = ∅ Again case (1) is controlled by k∇ ρθ k ∗ , dyadic , Q R by Lemma 8.2 so we only have todeal with case (2) and bound sup Q dyadic | ( ∇ ρθ ) Q | . Step 3.a: Case Q ⊂ Q R and l ( Q ) . Rηℓ .In this case Q touches the boundary of the support of ∇ ρ so we have the estimate k∇ ρ k L ∞ ( Q ) . l ( Q ) R since |∇ ρ | . /R . Also φ (0 , 0) = 0 and φ ∈ Lip(1 , / k ˜ φ ( x, t ) k L ∞ ( Q ) ≤ k φ ( x, t ) k L ∞ ( Q R ) . ℓR . Finally k x · ∇ ˜ φ Q r k L ∞ ( Q R ) . ℓR .Therefore | ( ∇ ρθ ) Q | ≤ k∇ ρ k L ∞ ( Q ) | θ | Q . l ( Q ) R | Q | ˆ Q | ˜ φ ( x, t ) − x · ∇ ˜ φ Q r | d x d t . l ( Q ) R ℓR . η. Step 3.b: Case Q ⊂ Q R and Rηℓ . l ( Q ) ≤ R .By the fundamental theorem of calculus we may write˜ φ ( x, t ) − ˜ φ (cid:18) r x | x | , t (cid:19) = x · ˆ r/ | x | ∇ ˜ φ ( λx, t ) d λ. ARABOLIC L p DIRICHLET PROBLEM AND VMO-TYPE DOMAINS 39 Therefore | ( ∇ ρθ ) Q | ≤ |∇ ρ || θ | Q = |∇ ρ | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ φ (cid:18) r x | x | , t (cid:19) + x · ˆ r/ | x | (cid:0) ∇ ˜ φ ( λx, t ) − ∇ ˜ φ Q r (cid:1) d λ + x · r | x | ∇ ˜ φ Q r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q . R (cid:13)(cid:13)(cid:13)(cid:13) ˜ φ (cid:18) r x | x | , t (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( Q ) + RR | Q | ˆ Q ˆ r/ | x | |∇ ˜ φ ( λx, t ) − ∇ ˜ φ Q r | d λ ! d x d t + ηRℓR . Since ˜ φ defined by (8.10) is tiled and reflected in time on cubes of scale r , and( rx/ | x | , ∈ Q r we control the first term above by1 R (cid:13)(cid:13)(cid:13)(cid:13) ˜ φ (cid:18) r x | x | , t (cid:19) − (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( Q ) ≤ R k φ − φ (0 , k L ∞ ( Q r ) . ℓrR . ℓη. Recall that r ∼ ηR , Rηℓ . l ( Q ) ≤ R and r ≤ | x | ∞ ≤ R so η/ ≤ λ ≤ 1. Weapply Fubini to the second term1 | Q | ˆ Q ˆ r/ | x | |∇ ˜ φ ( λx, t ) − ∇ ˜ φ Q r | d λ ! d x d t ≤ | Q | ˆ η/ ˆ Q |∇ ˜ φ ( λx, t ) − ∇ ˜ φ Q r | d x d t d λ. Let ˜ Q be the set formed by Q under the transformation ( x, t ) ( λx, t ). Wemay further cover ˜ Q by ∼ λ − translations of λQ with | λQ | / | ˜ Q | . λ . Thereforea similar proof to claim 8.3, using Lemma 8.2, gives1 | Q | ˆ Q |∇ ˜ φ ( λx, t ) − ∇ ˜ φ Q r | d x d t = 1 | ˜ Q | ˆ ˜ Q |∇ ˜ φ − ∇ ˜ φ Q r | . λ − λ | sQ | ˆ sQ |∇ ˜ φ − ∇ ˜ φ Q r | . η log (cid:18) rsl ( Q ) (cid:19) . η log (cid:18) ℓη (cid:19) . ε η − ε + η log(1 + ℓ )and hence after harmlessly integrating in λ we can control the second term by ˆ η/ η log (cid:18) ℓη (cid:19) d λ . ε η − ε + η log(1 + ℓ ) . Step 3.c: Case l ( Q ) ≥ R .As before in step 1.c, | ( ∇ ρθ ) Q | ≤ | ( ∇ ρθ ) Q R | , which can be further controlledby cubes that tile supp( ∇ ρ ). Therefore, this case is bounded as in step 3.b. Step 4: (8.13) holds; that is: k∇ ρθ k ∗ , dyadic . c ( η )Here we have 3 cases to consider:(1) Q ⊂ Q R (2) Q ⊂ R n \ supp( ∇ ρ )(3) Q R ⊂ Q Case (2) is obvious. Case (3) reduces down to case (1) by step 1.c, the reflectionand tiling of ˜ φ , and the supp( ∇ ρ ).Case (1): Using Lemma 8.4 this reduces down to showing that(a) | θ Q || Q | ˆ Q |∇ ρ − ( ∇ ρ ) Q | . c ( η )(b) 1 | Q | ˆ Q |∇ ρ ( θ − θ Q ) | . c ( η ) for Q dyadic and Q ⊂ Q R . Step 4.a: (a) holds for Q dyadic and Q ⊂ Q R . Step 4.a.i: Case Q ⊂ Q R and l ( Q ) . Rηℓ .By the naive bounds in step 3.a | θ | Q . ℓR . If we use the mean value theoremfor ∇ ρ similar to Step 2.2.b.i then1 | Q | ˆ Q |∇ ρ − ( ∇ ρ ) Q | . |∇ ρ | l ( Q ) . l ( Q ) R . Therefore | θ Q || Q | ˆ Q |∇ ρ − ( ∇ ρ ) Q | . ℓR l ( Q ) R . η. Step 4.a.ii: Case Q ⊂ Q R and Rηℓ . l ( Q ) ≤ R .Here we apply the same technique as step 3.b | θ Q || Q | ˆ Q |∇ ρ − ( ∇ ρ ) Q | ≤ | θ | Q |∇ ρ | . ε η − ε + η log(1 + ℓ ) . Step 4.b: (b) holds for Q dyadic and Q ⊂ Q R .1 | Q | ˆ Q |∇ ρ ( θ − θ Q ) | . R | Q | ˆ Q | θ − θ Q | . We split this into the now usual cases. Step 4.b.i: Case l ( Q ) . Rηℓ .By the intermediate and mean value theorems | ˜ φ − ˜ φ Q | . l ( Q ) ℓ and | x − x Q | . l ( Q ) so 1 R | Q | ˆ Q | θ − θ Q | = 1 R | Q | ˆ Q | ˜ φ − ˜ φ Q − x · ∇ ˜ φ Q r + ( x · ∇ ˜ φ Q r ) Q | . R l ( Q ) ℓ . η. Step 4.b.ii: Case Rηℓ . l ( Q ) < R .1 R | Q | ˆ Q | θ − θ Q | . R | θ | Q then applying the result from step 3.b gives1 | Q | ˆ Q |∇ ρ ( θ − θ Q ) | . ε η − ε + η log(1 + ℓ ) . Therefore by Lemma 8.1 we have shown ∇ Φ ∈ BMO( R n ) and the bound (8.21)holds. (cid:3) To finish proving Theorem 2.8 we need to establish property (iv). Lemma 8.6. Let Φ : R n − × R → R be defined in (8.11) with sup Q s = J s × I s ,Q s ⊂ Q d , s ≤ r | Q s | ˆ Q s ˆ I s | φ ( x, t ) − φ ( x, τ ) | | t − τ | d τ d t d x ≤ η (8.22) then Φ satisfies sup Q s = J s × I s | Q s | ˆ Q s ˆ I s | Φ( x, t ) − Φ( x, τ ) | | t − τ | d τ d t d x . η . (8.23) ARABOLIC L p DIRICHLET PROBLEM AND VMO-TYPE DOMAINS 41 Proof. Trivially since Φ is defined globallysup Q s = J s × I s | Q s | ˆ Q s ˆ I s | Φ( x, t ) − Φ( x, τ ) | | t − τ | d τ d t d x ≤ sup Q s = J s × I s | Q s | ˆ Q s ˆ I s | ˜ φ ( x, t ) − ˜ φ ( x, τ ) | | t − τ | d τ d t d x, where we interpret the value of ˜ φ where it is undefined as 0, i.e. ˜ φ ( x, t ) = 0 when( x, t ) supp( ˜ φ ). It remains to establishsup I s | I s | ˆ I s ˆ I s | ˜ φ ( x, t ) − ˜ φ ( x, τ ) | | t − τ | d τ d t d x . sup I s ⊂ I r | I s | ˆ I s ˆ I s | φ ( x, t ) − φ ( x, τ ) | | t − τ | d τ d t d x (8.24)pointwise in x , where Q r = J r × I r and is used to define Φ in (8.11). To simplifyour notation we drop the dependance on the spatial variables in ˜ φ and φ . We alsoset A := I s . Recall from (8.10) that˜ φ ( t ) = ( φ ( t ) t ∈ [ − r , r ] + 4 kr ,φ (2 r − t ) t ∈ [ r , r ] + 4 kr , for k ∈ Z . Let I k = [ − r , r ] + 4 kr and J k = [ r , r ] + 4 kr be intervals in timefor k ∈ Z . We partition A into disjoint pieces A = ∪ i I i ∪ j J j ∪ A ∪ A , where A and A are pieces that don’t contain either I i or J j .If A = A ∪ A we may as well assume (by translation an reflection) that A =[ a, r ], A = [ r , b ]. Let τ ′ , b ′ and A ′ be the images of τ, b and A respectivelyunder the map τ r − τ . Without loss of generality we only consider the case | A | > | A | . Since | t − τ | = | t − r | + | τ ′ − r | ≥ | t − τ ′ | we have for t ∈ A , τ ∈ A ˆ A ˆ A | ˜ φ ( t ) − ˜ φ ( τ ) | | t − τ | d τ d t = ˆ r a ˆ r b ′ | φ ( t ) − φ ( τ ′ ) | | t − (2 t − τ ′ ) | d τ ′ d t ≤ ˆ r a ˆ r b ′ | φ ( t ) − φ ( τ ′ ) | | t − τ ′ | d τ ′ d t ≤ ˆ A ˆ A | φ ( t ) − φ ( τ ′ ) | | t − τ ′ | d τ ′ d t. Therefore1 | A | ˆ A ˆ A | ˜ φ ( t ) − ˜ φ ( τ ) | | t − τ | d τ d t = 1 | A | (cid:18) ˆ A ˆ A +2 ˆ A ˆ A + ˆ A ˆ A (cid:19) | φ ( t ) − φ ( τ ′ ) | | t − τ ′ | d τ ′ d t . η . In the general case when A = ∪ i ∈I I i ∪ j ∈J J j ∪ A ∪ A we write the doubleintegral over A in terms of integrals X i,k ∈I ˆ I i ˆ I k | ˜ φ ( t ) − ˜ φ ( τ ) | | t − τ | d τ d t , X i ∈I ,j ∈J ˆ I i ˆ J j | ˜ φ ( t ) − ˜ φ ( τ ) | | t − τ | d τ d t and integrals that involve sets A or A or both (those are handled similar to theearlier calculation). Dealing with the first case, if i = k , t ∈ I i and τ ∈ I k then | t − τ | ∼ r | i − k | ; if i = k then | t − τ | = | t ′ − τ ′ | . Therefore X i,k ∈I ˆ I i ˆ I k | ˜ φ ( t ) − ˜ φ ( τ ) | | t − τ | d τ d t ∼ X i ∈I ˆ I ˆ I | φ ( t ) − φ ( τ ) | | t − τ | d τ d t + X i,k ∈I i = k r | i − k | ˆ I ˆ I | φ ( t ) − φ ( τ ) | d τ d t ≤ X i ∈I ˆ I ˆ I | φ ( t ) − φ ( τ ) | | t − τ | d τ d t + X i,k ∈I i = k | i − k | ˆ I ˆ I | φ ( t ) − φ ( τ ) | | t − τ | d τ d t . |I| ˆ I ˆ I | φ ( t ) − φ ( τ ) | | t − τ | d τ d t. In the second case X i ∈I ,j ∈J ˆ I i ˆ J j | ˜ φ ( t ) − ˜ φ ( τ ) | | t − τ | d τ d t . X i ∈I ,j ∈J| i − j |≤ ˆ I ˆ I | φ ( t ) − φ ( τ ) | | t − τ | + X i ∈I ,j ∈J| i − j |≥ r ( | i − j | − ˆ I ˆ I | φ ( t ) − φ ( τ ) | d τ d t . ( |I| + |J | ) ˆ I ˆ I | φ ( t ) − φ ( τ ) | | t − τ | d τ d t. Since | A | ∼ ( |I| + |J | ) | I | and I is one of the time intervals considered in thesupremum of (8.24)1 | A | ˆ A ˆ A | φ ( t ) − φ ( τ ) | | t − τ | d τ d t ∼ | I | ˆ I ˆ I | φ ( t ) − φ ( τ ) | | t − τ | d τ d t . η . (cid:3) References [Aro68] D. G. Aronson. 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School of Mathematics, The University of Edinburgh and Maxwell Institute ofMathematical Sciences, UK E-mail address : [email protected] School of Mathematics, The University of Edinburgh and Maxwell Institute ofMathematical Sciences, UK E-mail address : [email protected] Department of Mathematics, Yonsei University, Republic of Korea E-mail address ::