Parabolic Fractional Maximal Operator in Modified Parabolic Morrey Spaces
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.
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 .(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 . 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.
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
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 [10–14]). 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.
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 ) and Adams  studied boundedness of the Riesz potential for in Morrey spaces . Later on Chiarenza and Frasca  reproved boundedness of the Riesz potential in these spaces. By more general results of Guliyev  (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. ), whose isotropic counterpart was proved by Chiarenza and Frasca .
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  and Mazzucato  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 . The most systematic and general approach can certainly be found in the very recent book  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 ), 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, 34–39].
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.
Lemma 3.6. Let , and . Then and
Corollary 3.7. Let and . Then and
Corollary 3.8. Let , and . Then and
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 ). 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 .
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
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 .
(2) Sufficiency. Let . From (5.1) we have
In the case
In the case
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.
Thus Theorem 2.1<