#### Abstract

We study the continuity properties of the generalized fractional integral operator on the generalized local Morrey spaces and generalized Morrey spaces . We find conditions on the triple which ensure the Spanne-type boundedness of from one generalized local Morrey space to another , , and from to the weak space , . We also find conditions on the pair which ensure the Adams-type boundedness of from to for and from to for . In all cases the conditions for the boundedness of are given in terms of Zygmund-type integral inequalities on and , which do not assume any assumption on monotonicity of , , and in .

#### 1. Introduction

The theory of boundedness of classical operators of the real analysis, such as the maximal operator, Riesz potential, and the singular integral operators, from one weighted Lebesgue space to another one is well studied by now. Along with weighted Lebesgue spaces, Morrey-type spaces also play an important role in the theory of partial differential equations. Morrey spaces were first introduced by Morrey [1] in 1938 to study local behavior properties of the solutions of second-order elliptic partial differential equations. Furthermore, there are important applications for the theory of partial differential equations related to obtaining sharp a priori estimates and studying regularity properties of solutions in Morrey spaces. Recently, they proved to be useful also for the Navier-Stokes equations [2, 3]. However no attempt has been made to extend these results by using more generalized Morrey-type spaces. For example, sharp regularity properties of strong solutions to elliptic and parabolic equations with VMO coefficients in terms of general Morrey-type spaces are a good place to start the investigation.

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 . The fractional maximal operator and the Riesz potential are defined by If , then is the Hardy-Littlewood maximal operator.

For a measurable function the generalized Riesz potential is defined by for any suitable function on . If , , then we get the Riesz potential operator .

The generalized fractional integral operator was initially investigated in [4â€“6]. Nowadays many authors have been culminating important observations about especially in connection with Morrey spaces. Nakai [6] proved the boundedness of from the generalized Morrey spaces to the spaces for suitable functions , satisfying the doubling condition. The boundedness of from the generalized Morrey spaces to the spaces is studied by Eridani [7], Gunawan [8], Eridani et al. [9], Nakai [10], and Eridani et al. [11]. Guliyev [12] proved the Spanne- and Adams-type boundedness of in the spaces without any assumption on monotonicity of .

In this study, by using the method given by Guliyev in [13] (see also [12, 14]), we prove the Spanne-type boundedness of the operator from one generalized local Morrey space to another one , , and from to the weak space , . We also prove the Adams-type boundedness of the operator from generalized Morrey space to another one for and from to for .

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

#### 2. Generalized Local Morrey Spaces

We find it convenient to define the generalized Morrey spaces in the form as follows.

*Definition 1. *Let be a positive measurable function on and . We denote by the generalized Morrey space, the space of all functions with finite quasinorm: Also by we denote the weak generalized Morrey space of all functions for which

According to this definition, we recover the Morrey space and weak Morrey space under the choice :

*Definition 2. *Let be a positive measurable function on and . We denote by the generalized local (central) Morrey space, the space of all functions with finite quasinorm: Also by we denote the weak generalized local (central) Morrey space of all functions for which

*Definition 3. *Let be a positive measurable function on and . For any fixed we denote by the generalized local Morrey space, the space of all functions with finite quasinorm: Also by we denote the weak generalized local Morrey space of all functions for which

According to this definition, we recover the local Morrey space and weak local Morrey space under the choice : Furthermore, we have the following embeddings:

Wiener [15, 16] looked for a way to describe the behavior of a function at infinity. The conditions he considered are related to appropriate weighted spaces. Beurling [17] extended this idea and defined a pair of dual Banach spaces and , where . To be precise, is a Banach algebra with respect to the convolution, expressed as a union of certain weighted spaces; the space is expressed as the intersection of the corresponding weighted spaces. Feichtinger [18] observed that the space can be described bywhere is the characteristic function of the unit ball , is the characteristic function of the annulus , . By duality, the space , called Beurling algebra now, can be described by

Let and be the homogeneous versions of and by taking in (42) and (13) instead of there.

If or , then , where is the set of all functions equivalent to on . Note that and :

In order to study the relationship between central spaces and Morrey spaces, AlvĂˇrez et al. [19] introduced -central bounded mean oscillation spaces and central Morrey spaces , . If or , then . Note that and . Also define the weak central Morrey spaces .

The classical result by Hardy-Littlewood-Sobolev states that if , then the operator is bounded from to if and only if and for , the operator is bounded from to if and only if . Spanne and Adams studied boundedness of the Riesz potential in Morrey spaces. Their results can be summarized as follows.

Theorem 4 (Spanne, but published by Peetre [20]). *Let , , and . Moreover, let and . Then, for , the operator is bounded from to and, for , is bounded from to .*

Theorem 5 (Adams [21]). *Let , , , and . Then, for , the operator is bounded from to and, for , is bounded from to .*

Some authors [8, 12, 22â€“25] generalized Theorems 4 and 5 to generalized Morrey spaces and called them Spanne-type and Adams-type results for .

In [23] the following condition was imposed on :whenever , where does not depend on , and , jointly with the conditionwhere does not depend on and .

In [23] the following Spanne-type result was proved for on .

Theorem 6. *Let , , , and satisfy the conditions (15) and (16). Then the operator is bounded from to .*

The following Spanne-type result for on , containing results obtained in [23, 26], was proved in [12, 13] (see also [14]).

Theorem 7. *Let , , , , and satisfy the condition**where does not depend on and . Then the operator is bounded from to for and from to for .*

From Theorem 7 we get the following Spanne-type result for on .

Corollary 8. *Let , , , and satisfy the condition**where does not depend on and . Then the operator is bounded from to for and from to for .*

The following Spanne-type result for on , containing results obtained in [12], was proved in [27].

Theorem 9. *Let , , , and satisfy the condition**where does not depend on and . Then the operator is bounded from to for and from to for .*

#### 3. Some Weighted Inequalities

Let be a nonnegative function on . We denote by the space of all functions , , with finite norm and . Let be the set of all Lebesgue-measurable functions on and its subset consisting of all nonnegative functions on . We denote by the cone of all functions in which are nondecreasing on and

The following theorem is valid.

Theorem 10. *Let , be nonnegative measurable functions satisfying , , for any .**Then the identity operator is bounded from to on the cone if and only if*

*Proof. *If , are nonnegative functions on and is nondecreasing, then Also if , are nonnegative functions on and is nonincreasing, then Therefore for all where First we prove sufficiency. Assume that condition (22) holds. Then for all by (25).

To prove necessity assume that is bounded from to on the cone ; that is, where is independent of .

We note that for all and take . Observe that Also for all for sufficiently large hencefor all and condition (22) follows.

We will use the following statement on the boundedness of the weighted Hardy operator: where is a weight.

The following theorem was proved in [28] (see, also [29]).

Theorem 11. *Let , , and be weights on and let be bounded outside a neighborhood of the origin. The inequality**holds for some for all nonnegative and nondecreasing on if and only if**Moreover, the value is the best constant for (33).*

*Remark 12. *In (33) and (34) it is assumed that and .

#### 4. Spanne-Type Result for the Operator in

We assume thatso that the fractional integrals are well defined, at least for characteristic functions of complementary balls: In addition, we will also assume that satisfies the growth condition: there exist constants and such that

This condition is weaker than the usual doubling condition for the function : there exists a constant such that whenever and satisfy , and .

In the sequel for the generalized fractional integral operator we always assume that satisfies the conditions (37) and then denote the set of all such functions by . We will write, when ,

*Remark 13. *Typical examples of that we envisage are, for , and, for , The second one is used to control the Bessel potential (see also [30]).

The following theorem was proved in [11].

Theorem 14. *(1) Let . Then the operator is bounded from to if and only if there exists such that for all **(2) Let . Then the operator is bounded from to if and only if there exists such that for all *

The following lemma is valid.

Lemma 15. *Let and satisfy the conditions (35) and (37). If the condition (42) is fulfilled, then for the inequality **holds for any ball and for all .**If the condition (43) is fulfilled, then for the inequality**holds for any ball and for all .*

*Proof. *Let , , and . For arbitrary , set for the ball centered at and of radius . Write with and . Hence Since , and from condition (42) we get the boundedness of from to (see Theorem 14) and it follows that where constant is independent of .

It is clear that , implies . Then from conditions (35), (37) and by Fubiniâ€™s theorem we have Applying HĂ¶lderâ€™s inequality, we getMoreover, for all , the inequalityis valid. Thus Let . From the weak boundedness of and (43) it follows thatThen from (50) and (52) we get the inequality (45).

The following theorem is one of the main results of this paper.

Theorem 16. *Let , , and the function satisfy the conditions (35), (37), and (42). Let also satisfy the conditions**where does not depend on and . Then the operator is bounded from to for and from to for . Moreover, for ,**and, for ,*

*Proof. *By Lemma 15 and Theorems 10 and 11 we have, for ,and, for ,

Corollary 17. *Let , the function satisfies the conditions (35), (37), and (42). Let also satisfy the conditions**where does not depend on and . Then the operator is bounded from to for and from to for .*

In the case from Theorem 16 we get new Spanne-type result on generalized local Morrey spaces.

Corollary 18. *Let , , , and . Let also satisfy the condition**where does not depend on . Then the operator is bounded from to for and from to for .*

Also in the cases and , , from Theorem 16 we get local Morrey space variant of Theorem 4.

Corollary 19. *Let , , , and . Moreover, let and . Then, for , the operator is bounded from to and, for , is bounded from to .*

#### 5. Adams-Type Result for the Operator in

The following Adams-type result was proved in [31] (see also [12]).

Theorem 20. *Let , , and and let satisfy the conditions**where does not depend on and .**Then the operator is bounded from to for and from to for .*

The following Theorem was proved in [32].

Theorem 21. *Let and satisfies the condition**where does not depend on and . Then, for , the Hardy-Littlewood maximal operator is bounded from to and, for , is bounded from to .*

The following theorem is a main result of this paper on Adams-type estimate for generalized fractional integral operator . In the case we get Theorem 20 from this theorem.

Theorem 22. *Let , , and satisfy the conditions (37) and (42). Let also satisfy the condition (60) and**where does not depend on and .**Then the operator is bounded from to for and from to for .*

*Proof. *Let , , , , and . Write , where , , and . Then we have For , , following Hedbergâ€™s trick (see, e.g., [33, page 354]), we obtainFor , , from (49) we have