The Uniformly Parabolic Equations of Higher Order with Discontinuous Data in Generalized Morrey Spaces and Elliptic Equations in Unbounded Domains

We study the regularity of the solutions of the Cauchy-Dirichlet problem for linear uniformly parabolic equations of higher order with vanishing mean oscillation (VMO) coefficients. We prove continuity in generalized parabolic Morrey spaces M p , φ of sublinear operators generated by the parabolic Calderon-Zygmund operator and by the commutator of this operator with bounded mean oscillation (BMO) functions. We obtain strong solution belongs to the generalized Sobolev-Morrey space W m ,1 p , φ ∘ Q ð Þ . Also we consider elliptic equation in unbounded domains.


Introduction
We consider the higher order linear Cauchy-Dirichlet problem in Q ¼ Ω Â 0, T ð Þ, being a cylinder in  nþ1 , Ω ⊂ R n be a bounded domain 0 < T < ∞ where Þstands for the parabolic boundary of Q and D αβ ¼ , |α| ¼ P n k¼1 α k , β ¼ P n k¼1 β k . The unique strong solvability of this type problem was proved in [1]. In [2] the regularity of the solution in the Morrey spaces L p,λ  nþ1 À Á with p ∈ 1, ∞ ð Þ, λ ∈ 0, n þ 2 ð Þand also its Hölder regularity was studied. In [3] Nakai extend these studies on generalized Morrey spaces M p,φ  nþ1 À Á with a weight φ satisfying the integral condition ð ∞ r φ a, s ð Þ s ds ≤ cφ a, r ð Þ, ∀a ∈  nþ1 , r > 0: The generalized Morrey space is then defined to be the set of all f ∈ L p,loc  nþ1 À Á such that where the supremum is taken over all parabolic balls E with respect to the parabolic distance.
The main results connected with these spaces is the following celebrated lemma: let |Df | ∈ L p,nÀλ even locally, with n À λ < p, then u is Holder continuous of exponent α ¼ 1 À nÀλ p . This result has found many applications in theory elliptic and parabolic equations. In [2] showed boundedness of the maximal operator in L p,λ  nþ1 À Á that allows them to prove continuity in these spaces of some classical integral operators. So was put the beginning of the study of the generalized Morrey spaces M p,φ ,p > 1 with φ belonging to various classes of weight functions. In [3] proved boundedness of maximal and Calderon-Zygmund operators in M p,φ imposing suitable integral and doubling conditions on φ. These results allow to study the regularity of the solutions of various linear elliptic and parabolic value problems in M p,φ (see [4][5][6]). Here we consider a supremum condition on the weight which is optimal and ensure the boundedness of the maximal operator in M p,φ . We use maximal inequality to obtain the Calderon-Zygmund type estimate for the gradient of the solution of the problem (1) and (2) in the M p,φ .
The results presented here are a natural extension of the previous paper [7] to parabolic equations. Here we study the boundedness of the sublinear operators, generated by Calderon-Zygmund operators in generalized Morrey spaces and the regularity of the solutions of higher order uniformly elliptic boundary value problem in local generalized Morrey spaces where domain is bounded. Also hear we study higher order uniformly elliptic boundary value problem where domain is unbounded.
In paper [8] Byun, Palagachev and Wang is study the regularity problem for parabolic equation in classical Lebesgue classes and of Byun, Palagchev and Softova [9,10] where the problem studied in weighted Lebesgue and Orlicz spaces with a Muckenhoupt weight and the classical Morrey spaces L p,λ Q ð Þ with λ ∈ 0, n þ 2 ð Þ . In papers [11,12] the authors studied second order linear elliptic and parabolic equations with VMO coefficients.
Denote by a the coefficient a x, t ð Þ ¼ a αβ x, t ð Þ È É : Q ! M nÂn and by f x, t ð Þ nonhomogeneous term. Suppose that the operator is uniformly parabolic.
The paper is organized as follows. In section 2 we introduce some notations and give the definition of the generalized Morrey spaces M p,φ Q ð Þ. In section 3 we study sublinear operators generated by parabolic singular integrals in generalized Morrey spaces. In section 4 we is consider sublinear operators generated by non-singular integrals, in section 5 singular and non-singular integrals in generalized Morrey spaces. In section 6 we consider uniformly parabolic equations of higher order with VMO coefficients and proved regularity of solutions. In section 7 we study uniformly elliptic equations in unbounded domains.

Some notation and definition
The following notations are used in this paper: the letter C is used for various positive constants.
In the following, besides the standard parabolic metric ρ x, t ð Þ ¼ max jx 0 j, t j j 1 2 .
We use the equivalent one considered by Fabes and Riviere in [13]. The topology induced by ρ x, t ð Þconsists of the ellipsoids It is easy to see that the this metrics ore equivalent. In fact, for each E r there exist parabolic cylinders I and I with measure comparable to r nþ2 such that I ⊂ E r ⊂ I .
We give the definitions of the functional spaces that we are going to use. Let a ∈ L 1,loc  nþ1 À and let a E r ¼ E r j j À1 Ð E r a y ð Þdy be the mean value of the integral of a. Denote where E r ranges over all ellipsoids in  nþ1 . We say a ∈ BMO (bounded mean oscillation [14]) if We say a ∈ VMO (vanishing mean oscillation) [14] if a ∈ BMO and For any bounded cylinder Q we define BMO Q ð Þ and VMO Q ð Þ taking a ∈ L 1 Q ð Þ and Q r ¼ Q ∩ E r x ð Þ, x ∈ Q, instead of E r in the definition above. If a function a ∈ BMO or VMO, it is possible to extend the function in the whole of  nþ1 preserving its BMO-norm or VMO-modulus, respectively (see [15]). Any bounded uniformly continuous BUC Besides, BMO and VMO also contain discontinuous functions, and the following example shows the inclusion h : The generalized weak parabolic Morrey space WM 1,α R nþ1 ð Þconsists of all measurable functions such that We also define the space where ∂Q means the parabolic boundary Ω ∪ ∂Ω Â 0, T ð Þ ð Þ : In problem (1) and (1) and (2).

Sublinear operators generated by parabolic singular integrals in generalized Morrey spaces
Let f ∈ L 1  nþ1 À Á be a function with a compact support and a ∈ BMO: For any x ∉ supp f define the sublinear operators T and T a such that This operators are bounded in L p  nþ1 À Á satisfy the estimates where constants independent of a and f : Let we have the Hardy operator Hg r ð Þ ¼ 1 r Ð r 0 g s ð Þds, r > 0: Theorem 3.1. (see [12]) The inequality ess sup holds for all non-increasing functions g : Lemma 3.1. (see [12] and let T be a sublinear operator satisfying (6). i. If p > 1 and T is bounded on L p  nþ1 À Á , then ii. If p ¼ 1 and T is bounded from where the constants are independent of r, x 0 and f : Theorem 3.2. (see [12]) Let p ∈ 1, ∞ ½ Þand φ x, r ð Þbe a measurable positive function satisfying and let T be a sublinear operator satisfying (6).
i. If p > 1 and T is bounded on L p  nþ1 À Á , then T is bounded on M p,φ  nþ1 À Á , and ii. If p ¼ 1 and T is bounded from with constants independent of f : Our next step is to show boundedness of T a in M p,φ  nþ1 À Á : For this we recall some properties of the BMO functions.  As an immediate consequence of (7) we get the following property. Corollary 3.1. Let a ∈ BMO: Then, for all 0 < 2r < s, Now we estimate the norm of T a : Lemma 3.3. (see [12]) Let a ∈ BMO: and T a be a bounded operator in L p  nþ1 À Á , p ∈ 1, ∞ ð Þ, satisfying (7) and (8). Suppose that, for any f ∈ L p,loc  nþ1 Then, where C is independent of a, f , x 0 and r:

is independent of x and r: Suppose that a ∈ BMO and let T a be a sublinear operator satisfying (7). If T a is bounded in L
and

and T, T a be sublinear operators such that
Let both the operators be bounded in L p D nþ1 þ À Á , satisfy the estimates constants C independent of a and f : The following results hold, which can be proved in the some manner as in Section 3 (see [12]). Lemma 4.1. Let f ∈ L p,loc D nþ1 where the constant c is independent of r, x 0 and f : Theorem 4.1. Suppose φ be a weight function satisfying (14), and let T be a sublinear operator satisfying (22) and (24). Then T is bounded in M p,φ D nþ1 with a constant c independent of f : and T a satisfy (23) and (24). Suppose that, for all f ∈ L p,loc D nþ1 Then with a constant c independent of a, f , x 0 and r.
Þbe a weight function satisfying (20) and T a be a sublinear operator satisfying (7), (8). Then sublinear operator T a is bounded in M p,φ D nþ1 constant c independent of a and f .

Singular and non-singular integrals in generalized Morrey spaces
We apply the above results to Calderon-Zygmund-type operators with parabolic kernel. Since these operators are sublinear and bounded in L p  nþ1 À Á , their continuity in M p,φ follows immediately. We are called a parabolic Calderon-Zygmund kernel if the following a measurable function K x, ξ ð Þ :  nþ1 Â  nþ1 n 0 f g ! R.
1. K x, Á ð Þ is a parabolic Calderon-Zygmund kernel for a.e. x ∈  nþ1 : which means the singular integrals ∥C a, f constant c independent of a and f . We define the extensions and then the singular integral satisfying inequalities Corollary 5.2. Let a ∈ VMO. Then for any ε > 0 there exists a positive number r 0 ¼ r 0 ε, η a ð Þsuch that for any E r x 0 ð Þ with a radius r ∈ 0, where c is independent of E, f , r, and x 0 : For the proof of corollary see [12].
For any x 0 ∈ R n þ and any fixed t > 0, define the generalized reflexion where |α| ≤ m, |β| ≤ m, a n αβ x ð Þ is the last row of the coefficients matrix a x ð Þ ¼ a αβ x ð Þ À Á of (1). The function τ 0 x ð Þ maps R n þ into R n À , and the kernel þ , there exists positive constants K 1 and K 2 such that Since K x, τ x ð Þ À y ð Þis still homogeneous and satisfies 1 b , we have Hence, the operators (36) are sublinear and bounded in L p D nþ1 constant C independent of a and f . Corollary 5.3. For any a ∈ VMO. Then there exists a positive number r 0 ¼ r 0 ε, φ a ð Þ such that for any E r x 0 ð Þ with a radius r ∈ 0, r 0 ð Þand all ∥f ∥ M p,φ E þ where C is independent of E, f , r and x 0 , ε > 0.

Proof of the first main result
Now using boundedness of singular integral of Calderon-Zygmund operators in generalized Morrey spaces we will get interval estimates for solutions of problem (1), (2) with coefficients from VMO spaces.
Let Ω to be open bounded domain in R n , n ≥ 3 and we suppose that its boundary is sufficiently smoothness.
Now we give boundary estimates. For any fixed x 0 , r ð Þ∈  nþ1 Â R þ define the semicylinders C þ