Parabolic oblique derivative problem with discontinuous coefficients in generalized weighted Morrey spaces

Abstract We obtain the global weighted Morrey-type regularity of the solution of the regular oblique derivative problem for linear uniformly parabolic operators with VMO coefficients. We show that if the right-hand side of the parabolic equation belongs to certain generalized weighted Morrey space Mp,ϕ(Q, w), than the strong solution belongs to the generalized weighted Sobolev- Morrey space W˙2,1p,φ(Q,ω)$\dot W_{2,1}^{p,\varphi }\left( {Q,\omega } \right)$.


Introduction
We consider the regular oblique derivative problem in generalized weighted Sobolev-Morrey space P W p;' 2;1 .Q; !/ for linear non-divergence form parabolic equations in a cylinder The unique strong solvability of this problem was proved in [38]. In [39] Softova studied the regularity of the solution in the Morrey spaces L p; with p 2 .1; 1/, 2 .0; n C 2/ and also its Hölder regularity. In [41] Softova extended these studies on generalized Morrey spaces L p;' with a Morrey function ' satisfying the doubling and integral conditions introduced in [27,31]. The approach associated to the names of Calderón and Zygmund and developed by Chiarenza, Frasca and Longo in [7,8] consists of obtaining of explicit representation formula for the higher order derivatives of the solution by singular and nonsingular integrals. Further the regularity properties of the solution follows by the continuity properties of these integrals in the corresponding spaces. In [39] and [40] the regularity of the corresponding operators in the Morrey and generalized Morrey spaces is studied, while in [38] we can find the corresponding results obtained in L p by [9] and [5]. In recent works there have been studied the regularity of the solutions of elliptic and parabolic problems with Dirichlet data on the boundary in generalized Morrey spaces M p;' with a weight ' satisfying (10) with w Á 1 (cf. [18,19]). Precisely, a boundedness in M p;' was obtained for sub-linear operators generated by singular integrals as the Calderon-Zygmund. More results concerning sub-linear operators in generalized Morrey spaces can be found in [3,12,40] see also the references therein. After studying generalized Morrey spaces in detail, researchers passed to weighted Morrey spaces and generalized weighted Morrey spaces. Recently, Komori and Shirai [23] defined the weighted Morrey spaces L p;Ä .w/ and studied the boundedness of some classical operators, such as the Hardy-Littlewood maximal operator or the Calderón-Zygmund operator on these spaces. Also, Guliyev in [13] first introduced the generalized weighted Morrey spaces M p;' w and studied the boundedness of the sublinear operators and their higher order commutators generated by Calderón-Zygmund operators and Riesz potentials in these spaces (see, also [15,17]). Note that, Guliyev [13] gave the concept of generalized weighted Morrey space which could be viewed as an extension of both M p;' and L p;Ä .w/.
We call weight a positive measurable function defined on R n R C : In [29] Muckenhoupt shows that the maximal inequality holds in weighted Lebesgue spaces L q w if and only if the weight w satisfies the following integral condition called parabolic Muckenhoupt condition or parabolic A q -condition. We say that the measurable, nonnegative function w W R n ! R C satisfies the parabolic A q -condition for q 2 .1; 1/ if for all parabolic cylinders I in R nC1 . Then w.I/ means the weighted measure of I; that is This measure satisfies strong and reverse doubling property. Precisely, for each I and each measurable subset A I, there exist positive constants c 1 and 1 2 .0; 1/ such that where c 1 and 1 depend on n and q but not on I and A. Throughout this paper the following notations are to be used: x D .x 0 ; t / D .x 00 ; x n ; t / 2 R nC1 , R nC1 C D fx 0 2 R n ; t > 0g and D nC1 C D fx 00 2 R n 1 ; x n > 0; t > 0g, D i u D @u=@x i , D ij u D @ 2 u=@x i @x j , D t u D u t D @u=@t stand for the corresponding derivatives while Du D .D 1 u; : : : ; D n u/ and D 2 u D fD ij ug n i;j D1 mean the spatial gradient and the Hessian matrix of u. For any measurable function f and A R nC1 we write where jAj is the Lebesgue measure of A. Through all the paper the standard summation convention on repeated upper and lower indexes is adopted. The letter C is used for various constants and may change from one occurrence to another.

Definitions and statement of the problem
Let R n , n 1 be a bounded C 1;1 -domain, Q D .0; T / be a cylinder in R nC1 C , and S D @ .0; T / stands for the lateral boundary of Q. We consider the problem under the following conditions: (i) The operator B is supposed to be uniformly parabolic, i.e. there exists a constant ƒ > 0 such that for almost all x 2 Q ( ƒ 1 j j 2 Ä a ij .x/ i j Ä ƒj j 2 ; 8 2 R n ; a ij .x/ D a j i .x/; i; j D 1; : : : ; n: The symmetry of the coefficient matrix a D fa ij g n i;j D1 implies essential boundedness of a ij 's and we set (ii) The boundary operator B is prescribed in terms of a directional derivative with respect to the unit vector field l.x/ D .l 1 .x/; : : : ; l n .x//, x 2 S. We suppose that B is a regular oblique derivative operator, i.e., the field l is never tangential to S: Here Lip. N S/ is the class of uniformly Lipschitz continuous functions on N S and n.x/ stands for the unit outward normal to @ .
In the following, besides the parabolic metric %.x/ D max.jx 0 j; jt j 1=2 / and the defined by it parabolic cylinders W jx 0 y 0 j < r; jt j < r 2 g; jI r j D C r nC2 : we use the equivalent one .x/ D jx 0 j 2 C p jx 0 j 4 C4t 2 2 Á 1 2 (see [9]). The balls with respect to this metric are ellipsoids Because of the equivalence of the metrics all estimates obtained over ellipsoids hold true also over parabolic cylinders and in the following we shall use this without explicit references. jf .y/ f E r jdy; for every R > 0 where E r ranges over all ellipsoids in R nC1 . The Banach space BMO (bounded mean oscillation) consists of functions for which the following norm is finite A function a belongs to VMO (vanishing mean oscillation) with VMO-modulus Á a .R/ provided For any bounded cylinder Q we define BMO.Q/ and VMO.Q/ taking a 2 L 1 .Q/ and Q r D Q \ I r instead of E r in the definition above.
According to [1,21] having a function a 2 BMO=VMO.Q/ it is possible to extend it in the whole R nC1 preserving its BMO-norm or VMO-modulus, respectively. In the following we use this property without explicit references. For this goal we recall some well known properties of the BMO functions.
Lemma 2.2 (John-Nirenberg lemma, [20]). Let a 2 BMO and p 2 .1; 1/. Then for any ball B there holds 0 @ 1 jBj As an immediate consequence of Lemma 2.2 we get the following property.
where the constant is independent of a; x; t and r: As mentioned before, we call weight a positive measurable function defined on R n R C . Given a weight w and a measurable set S we denote by where 1 p C 1 p 0 D 1. Note that, for any ball we have (see [11]) and OEw A 1 is the smallest A for which the above inequality holds. It is an immediate consequence of (7) that whenever w 2 A p than it satisfies the doubling property, precisely w.2B r / Ä C.n; p/w.B r / : The following lemma collects some of the most important properties of the Muckenhoupt weights. 11]). We have the following: (1) If w 2 A p for some 1 Ä p < 1, then for all > 1 we have (2) The following equality is valid:  (1) there exist constants C 1 , C 2 > 0, such that for allˇ> 0˚x The space M p;' .Q; w/ consists of L p ! .Q/ functions provided the following norm is finite kf k p;';wIQ D sup x2R n ;r>0 with C D C.n; p; OE! A p ; ƒ; @ ; T; kak 1IQ ; Á a / and Á a D If '.x; r/ D r . n 2/=p , then M p;' Á L p; and the condition (10) holds with a constant depending on n, p and . If '.x; r/ D !.x; r/ 1=p r .nC2/=p with ! W R nC1 R C ! R C satisfying the conditions than we obtain the spaces L p;! studied in [27,31]. The following results are obtained in [19] and treat continuity in M p;' .R nC1 ; !/ of certain singular and nonsingular integrals.
where C is independent of ", f , r and x 0 .
For any x 0 2 R n C and any fixed t > 0 define the generalized reflection T .x/ D .T 0 .x/; t /; T 0 .x/ D x 0 2x n a n .x 0 ; t / a nn .x 0 ; t / where a n .x/ is the last row of the coefficients matrix a.x/ of (3). The function T 0 .x/ maps R n C into R n and the kernel K.xI T .x/ y/ D K.xI T 0 .x/ y 0 ; t / is a nonsingular one for any x; y 2 D nC1 C . Taking Q x D .x 00 ; x n ; t / there exist positive constants k 1 and k 2 such that with a constant independend of a and f .
where C is independent of ", f , r and x 0 .

Proof of the main result
As it follows by [39], the problem (3) is uniquely solvable in P W p 2;1 .Q; !/. We are going to show that f 2 M p;' .Q; !/ implies u 2 P W p;' 2;1 .Q; !/. For this goal we obtain an a priori estimate of u. Following the method used by Chiarenza, Frasca and Longo in [7] and [8], we prove the results considering two steps.
Interior estimate. For any x 0 2 R nC1 C consider the parabolic semi-cylinders C r .x 0 / D B r .
where . 1 ; : : : ; nC1 / is the outward normal to S n . Here .xI / is the fundamental solution of the operator B and .xI / D @ 2 .xI /=@ i @ j . Because of density arguments the representation formula (21) still holds for any v 2 W p 2;1 .C r .x 0 /; !/. The properties of the fundamental solution (cf. [5,25,38]) imply ij are Calderon-Zygmund kernels in the sense of Definition 2.9. We denote by K ij and C ij the singular integrals defined in (12) with kernels K.xI x y/ D ij .xI x y/. Then we can write that Because of Corollaries 2.11 and 2.12 and the equivalence of the metrics we get  and taking the supremo with respect to Â and Â 0 we get The interpolation inequality [26,Lemma 4.2] gives that there exists a positive constant C independent of r such that where C depends on n, p, OE! 1 2 A p , ƒ, T , kDk 1IQ , Á a .r/, kak 1;Q and d i st . 0 ; @ 00 /. Boundary estimates. For any fixed R > 0 and x 0 D .x 00 ; 0; 0/ define the semi-cylinders Without lost of generality we can take x 0 D .0; 0; 0/. Define B C R D fjx 0 j < R; x n > 0g, S C R D fjx 00 j < R; x n D 0; t 2 .0; R 2 /g and consider the problem Let u 2 W p 2;1 .C C R ; !/ with u D 0 for t Ä 0 and x n Ä 0, then the following representation formula holds (see [26,38]) where I ij .x/ D P:V: .xI x y/F .xI y/dy C f .x/ Z S n j .xI y/ i d y ; i; j D 1; : : : ; nI H ij .x/ D .G ij 2 g/.x/ C g.x 00 ; t / Z S n G j .xI y 00 ; x n ; /n i d .y 00 ; /; i; j D 1; : : : ; n; @T .x/ @x n D 2 a n1 .x/ a nn .x/ ; : : : ; 2 a nn 1 .x/ a nn .x/ ; 1 Á : Here the kernel G D Q, is a byproduct of the fundamental solution and a bounded regular function Q. Hence its derivatives G ij behave as ij and the convolution that appears in H ij is defined as follows .G ij 2 g/.x/ D P:V: .xI x 00 y 00 ; x n ; t /g.y 00 ; 0; /dy 00 d ; g.x 00 ; 0; t / D OE.l k .0/ l k .x 00 ; 0; t //D k u l k .0/. k F / j x n D0 .x 00 ; 0; t /; .
.xI x y/F .xI y/dy: Here I ij are a sum of singular integrals and bounded surface integrals hence the estimates obtained in Corollaries 2.11 and 2.12 hold true. On the nonsingular integrals J ij we apply the estimates obtained in Theorem 2.13 and Corollary 2.14 that give for all i; j D 1; : : : ; n. To estimate the norm of H ij we suppose that the vector field l is extended in C C R preserving its Lipschitz regularity and the norm. This automatically leads to extension of the function g in C C R that is  The convolution k f is a Riesz potential. On the other hand Finally unifying (26), (28) and (29) Making a covering fC C g,˛2 A such that Q n Q 0 S 2A C C , considering a partition of unity subordinated to that covering and applying (30) for each C C we get kuk W p;' 2;1 .QnQ 0 ;!/ Ä C kf k p;';!IQ (31) with a constant depending on n, p, OE! 1 2 A p , ƒ, T , d i am , kDk 1IQ , Á a , kak 1IQ , klk Lip. N S/ , and kDlk 1;S . The estimate (11) follows from (24) and (31).