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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 543475, 20 pages
http://dx.doi.org/10.1155/2012/543475
Research Article

Parabolic Fractional Maximal Operator in Modified Parabolic Morrey Spaces

1Department of Mathematics, Ahi Evran University, Kirsehir, Turkey
2Institute of Mathematics and Mechanics, Baku State University, Baku, Azerbaijan
3Baku State University, 1148 Baku, Azerbaijan

Received 30 August 2012; Revised 30 September 2012; Accepted 1 October 2012

Academic Editor: Dachun Yang

Copyright © 2012 Vagif S. Guliyev and Kamala R. Rahimova. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove that the parabolic fractional maximal operator , , is bounded from the modified parabolic Morrey space to the weak modified parabolic Morrey space if and only if and from to if and only if . Here is the homogeneous dimension on . In the limiting case we prove that the operator is bounded from to . As an application, we prove the boundedness of from the parabolic Besov-modified Morrey spaces to . As other applications, we establish the boundedness of some Schrödinger-ype operators on modified parabolic Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.

1. Introduction

The theory of boundedness of classical operators of the real analysis, such as the maximal operator, the fractional maximal operators, the fractional integral operators, and the singular integral operators, from one weighted Lebesgue space to another one is well studied by now. These results have good applications in the theory of partial differential equations. However, in the theory of partial differential equations, along with Morrey spaces, modified Morrey spaces also play an important role (see [1, 2]).

For and , we denote by the open ball centered at of radius and by denote its complement. Let be the Lebesgue measure of the ball .

Let be a real matrix, all of whose eigenvalues have positive real part. Let and set . Then, there exists a quasi-distance associated with such that(a), for every ;(b), and;(c), where , , and is a measure on the ellipsoid .

Then, becomes a space of homogeneous type in the sense of Coifman-Weiss. Moreover, we always assume the following properties on .(d) For every ,

Here and are some positive constants. Similar properties hold for which is associated with the matrix . Here is the adjoint matrix of .

There are some important examples for the above spaces.(1)Let . In this case, is defined by the unique solution of , and . This space is just the one studied by Calderón and Torchinsky in [3].(2)Let be a diagonal matrix with positive diagonal entries and let , be the unique solution of .(a) If all diagonal entries are greater than or equal to , this space was studied by Fabes and Rivière [4]. More precisely they studied the weak and estimates of the singular integral operators on this space in 1966.(b)If there are diagonal entries smaller than , then satisfies the above (a)–(d) with .

Thus , endowed with the metric , defines a homogeneous metric space [4, 5]. The balls with respect to , centered at of radius , are just the ellipsoids , with the Lebesgue measure , where is the volume of the unit ellipsoid in . Let also be the complement of . If , then clearly and . Note that in the standard parabolic case we have Let . The parabolic fractional maximal function and the parabolic fractional integral are defined by

If , then is the parabolic maximal operator. If , then is the fractional maximal operator, is the Hardy-Littlewood maximal operator and is the Riesz potential.

It is well known that the fractional maximal operator, the fractional integral operator, and Calderón-Zygmund operators play an important role in harmonic analysis (see [69]).

Definition 1.1. Let , , . We denote by the parabolic Morrey space, and by the modified parabolic Morrey space the set of locally integrable functions , , with the finite norms respectively.
Note that and if or , then , where the symbol means continuous embedding (let be the normed spaces, then by definition means that there exists such that for all and is the set of all functions equivalent to on .

Definition 1.2 (see [1014]). Let , . We denote by the weak parabolic Morrey space and by the modified weak parabolic Morrey space the set of locally integrable functions with finite norms: respectively.

Note that If , then is the classical Morrey spaces [15] and is the modified Morrey spaces [2].

Note that the parabolic generalized Morrey spaces are defined as follows (see, e.g., [1618], etc.)

Definition 1.3. Let be a positive measurable function on and . We denote by the parabolic generalized Morrey space, the space of all functions with finite quasinorm: Notice that if we let , then we obtain the modified Morrey norm.
The anisotropic result by Hardy-Littlewood-Sobolev states that if , then is bounded from to if and only if and for is bounded from to if and only if . Spanne (see [19]) and Adams [1] studied boundedness of the Riesz potential for in Morrey spaces . Later on Chiarenza and Frasca [20] reproved boundedness of the Riesz potential in these spaces. By more general results of Guliyev [21] (see also [17, 18, 22, 23]) one can obtain the following generalization of the results in [1, 19, 20] to the anisotropic case.

Theorem A. Let and , .(1)If , then the condition is necessary and sufficient for the boundedness of the operator from to .(2)If , then the condition is necessary and sufficient for the boundedness of the operator from to .

If , then and the statement of Theorem A reduces to the aforementioned result by anisotropic version of Hardy-Littlewood-Sobolev.

Recall that, for , hence Theorem A also implies the boundedness of the fractional maximal operator . It is known that the parabolic maximal operator is also bounded on for all and (see, e.g. [24]), whose isotropic counterpart was proved by Chiarenza and Frasca [20].

In this paper we study the parabolic fractional maximal integral in the modified parabolic Morrey space . In the case we prove that the operator is bounded from to if and only if, . In the case we prove that the operator is bounded from to if and only if, . In the limiting case we prove that the operator is bounded from to .

The structure of the paper is as follows. In Section 1 the boundedness of the maximal operator in modified Morrey space is proved. The main result of the paper is the Hardy-Littlewood-Sobolev inequality in modified parabolic Morrey space for the parabolic fractional maximal operator, established in Section 2. In Section 3 by using the boundedness of the parabolic fractional maximal operators we establish the boundedness of some Schrödinger type operators on modified Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.

By we mean that with some positive constant independent of appropriate quantities. If and , we write and say that and are equivalent.

2. Main Results

Theorem 2.1. Let , , and .(1)If , then the condition is necessary and sufficient for the boundedness of the operator from to .(2)If , then the condition is necessary and sufficient for the boundedness of the operator from to .(3)If , then the operator is bounded from to .

Besov-Morrey (and Triebel-Lizorkin-Morrey) spaces attracted some attention in these two decades. Kozono and Yamazaki [25] and Mazzucato [26] used these spaces in the theory of Navier-Stokes equations. Some properties of the spaces including the wavelet characterizations were described in the papers by Sawano [27, 28], Sawano and Tanaka [29, 30], Tang and Xu [31]. The most systematic and general approach can certainly be found in the very recent book [32] of Yuan et al., we also recommend this monograph for further up-to-date references on this subject.

In the following theorem we prove the boundedness of in the parabolic Besov-modified Morrey spaces on (see [33]), whose norm is given by where , and .

These spaces generalize certain Besov-Morrey and Triebel-Lizorkin-Morrey spaces. As a general theory of Besov-Triebel-Lizorkin spaces, the Besov-Morrey and Triebel-Lizorkin-Morrey spaces are introduced due to the study of Navier-Stokes equations and attract some attention in recent years. Another scales of generalized Besov and Triebel-Lizorkin spaces, the Besov-type space and Triebel-Lizorkin-type space, were introduced by Yang and Yuan in [34, 35] and proved therein to be closely related to the theory of spaces. For further developments and applications of these spaces, we also refer to [32, 3439].

Theorem 2.2. Let , and . If , , and , then the operator is bounded from the space to . More precisely, there is a constant such that holds for all .

3. Some Several Embeddings

Lemma 3.1. Let , . Then

Proof. Let . Then from (1.5) we have that , and , .
Let now . Then Therefore, and the embedding is valid.
Thus and .

The following statement can be proved analogously.

Lemma 3.2. Let , , then

Lemma 3.3. Let , and . Then

Proof. Let , , . By the Hölder's inequality we have Moreover, therefore and

Lemma 3.4. Let , . Then for .

Proof. Let , , , and . By the Hölder's inequality we have therefore and

For the we define the following fractional maximal functions: In the case we denote is simply denoted by .

Lemma 3.5. Let , and . Then and

Proof. We have the following.

From Lemmas 3.1 and 3.5 we get the following.

Lemma 3.6. Let , and . Then and

In the case from Lemmas 3.5 and 3.6 one gets that for the the following property is valid.

Corollary 3.7. Let and . Then and

In the case from Lemmas 3.3 and 3.5 we get for the the following property is valid.

Corollary 3.8. Let , and . Then and

From Lemmas 3.4 and 3.6 one gets that for the following property is valid.

Corollary 3.9. Let , and . Then for and

4. Boundedness of the Parabolic Maximal Operator

In this section we study the boundedness of the maximal operator .

Theorem 4.1 (see [24]). If , , then and where depends only on and .
If , , , then and where depends only on , , , and .

Applying Theorem 4.1, one obtains the following result.

Theorem 4.2. If , , then and where depends only on and .
If , , , then and where depends only on , , , and .

Proof. It is obvious that (see Lemmas 3.1 and 3.2) for and for .
Let . By the boundedness of on (see, e.g. [3]) and from Theorem 4.1 we get

Let . By the boundedness of from to (see, e.g., [3]) and from Theorem 4.1 we have

5. Proof of Main Results

Proof of Theorem 2.1. (1) Sufficiency. Let , , , , and . Then

For we get

By the Hölder inequality

Thus for all

Minimizing with respect to , at

we have

Then

Hence, by Theorem 4.2, we have

which implies that is bounded from to .

Necessity. Let and assume that is bounded from to .

Define , . Then

By the boundedness of from to

If , then by letting we have for all .

As well as if , then at we obtain for all .

Therefore .

(2) Sufficiency. Let . From (5.1) we have

Then

Taking into account inequality (5.2) and Theorem 4.2, we have where and thus if , then and consequently, .

In the case

then

In the case

then

Finally we have

Necessity. Let be bounded from to . We have

By the boundedness of from to where depends only on , , and .

If , then by letting we have for all , which is impossible.

Similarly, if , then for we obtain for all , which is impossible.

Therefore .

Thus Theorem 2.1 is proved.

Proof of Theorem 2.2. By the definition of the parabolic Besov-modified Morrey spaces on it suffices to show that where .
It is easy to see that commutes with , that is, . Hence we obtain Taking -norm on both sides of the last inequality, we obtain the desired result by using the boundedness of from to .
Thus the proof of the Theorem 2.2 is completed.

6. Parabolic Schrödinger-Type Operators and

In this section we consider the parabolic Schrödinger operator where is a nonnegative potential which belongs to the parabolic reverse Hölder class . Examples of such potentials are all positive polynomials but also singular functions like for . We prove the modified parabolic Morrey space estimates for the operators and , where .

The investigation of Schrödinger operators on the Euclidean space with nonnegative potentials which belong to the reverse Hölder class has attracted attention of a number of authors (cf. [4042]). Shen [41] studied the Schrödinger operator , assuming that the nonnegative potential belongs to the reverse Hölder class for and he proved the boundedness of the operators , , , and . Kurata and Sugano generalized Shen’s results to uniformly elliptic operators in [43]. Sugano [44] also extended some results of Shen to the operator , and , and . Following Shen's approach, Gao and Jiang extend the results to the parabolic case. In [45], they consider the parabolic operator where is a nonnegative potential depending only on the space variables and, under the assumptions and , they proved the boundedness of in .

The main purpose of this section is to investigate the modified parabolic Morrey space boundedness of the operators Note that the operator in [45] is the special case of .

It is worth pointing out that we need to establish pointwise estimates for , and their adjoint operators by using the estimates of fundamental solution for the Schrödinger operator on in [45]. And we prove the modified parabolic Morrey space boundedness of the parabolic fractional maximal operators.

Definition 6.1. (1) A nonnegative locally integrable function on is said to belong to the parabolic reverse Hölder class if there exists such that the reverse Hölder inequality holds for every parabolic cylinder of center and radius in .
(2) Let . We say , if there exists a constant such that holds for every parabolic cylinder in .

Clearly, for . But it is important that the class has a property of “self-improvement”, that is, if , then for some (see [46]).

By the functional calculus, we may write, for all ,

Let . From it follows that where

The following two pointwise estimates for and which were proved in [42], Lemma 3.2 with the potential .

Theorem A. Suppose and . Then, for any where .

Theorem B. Suppose , and . Then, for any where .

Note that the similar estimates for the adjoint operators and with the potential for some are also valid (see [47]).

Theorem C. Suppose for some , and let . Then there exists a constant such that where .

Theorem D. Suppose for some , , and . And let

Then there exists a constant such that where .

The above theorems will yield the modified parabolic Morrey estimates for and .

Corollary 6.2. Assume that , and . Let , , and

Then, for any <