Research Article | Open Access
Weighted Hardy-Type Inequalities in Variable Exponent Morrey-Type Spaces
We study the boundedness of weighted multidimensional Hardy-type operators and of variable order , with radial weight , from a variable exponent locally generalized Morrey space to another . The exponents are assumed to satisfy the decay condition at the origin and infinity. We construct certain functions, defined by , , and , the belongness of which to the resulting space is sufficient for such a boundedness. Under additional assumptions on , this condition is also necessary. We also give the boundedness conditions in terms of Zygmund-type integral inequalities for the functions and .
Influenced by various applications, for instance, mechanics of the continuum medium and variational problems, in the last two decades the study of various mathematical problems in the spaces with nonstandard growth attracts the attention of researchers in various fields. This notion relates first of all to the generalized Lebesgue spaces , , known also as Lebesgue spaces with variable exponent . We refer to the existing books [1–3] in the field.
This variable exponent boom naturally touched Morrey spaces. Morrey spaces (with constant exponents) in its classical version were introduced in  in relation to the study of partial differential equations and presented in various books; see, for example, [5–7]; we refer also to a recent overview of Morrey spaces in , where various generalizations of Morrey spaces may be also found.
They were widely investigated during the last decades, including the study of classical operators of harmonic analysis, maximal, singular, and potential operators on Morrey spaces, and their generalizations were studied. We refer for instance to papers [9–16] and the references therein; in particular, Hardy operators in Morrey type spaces with constant were studied in [17–20].
The Morrey spaces with variable exponents and were introduced and studied in [21–23]. Generalized Morrey spaces , with variable exponents were studied in ; see also another version of Morrey-type spaces in ; we also refer to  for the so-called complementary Morrey spaces of variable order in the spirit of ideas of .
In the above cited paper maximal, singular, and potential operators were studied. This paper seems to be the first one where Hardy-type integral inequalities are studied in Morrey-type spaces with variable exponents. Concerning Hardy-type inequalities and related problems and applications, we refer to the books [27, 28].
The paper is organized as follows. In Section 2, we give necessary preliminaries on variable exponent Lebesgue spaces. In Section 3, we define our main object-variable exponent Morrey spaces and prove important weighted estimates of functions in Morrey spaces; see Theorem 10.
By means of these estimates in Section 4, we prove our main statements for Hardy operators in variable exponent generalized Morrey spaces. We also consider the necessity of the obtained conditions. In Section 4.4, under some additional assumptions on we obtain the boundedness conditions in a different form via Zygmund-type conditions on and provide a direct relation between and .
In Theorem 18 of this section, we specially single out the nonweighted case where we show that the Hardy inequalities in Morrey spaces are completely determined by the values and .
In the appendix we recall some notions related to the Bary-Zygmund-Stechkin class and Matuszewska-Orlicz indices which sporadically are used in the paper.
2. Preliminaries on Variable Exponent Lebesgue Spaces
We first recall the basic definitions related to variable exponent spaces. By , we always denote an open set in , , and , . Let also and .
Let be a measurable function on with values in . We suppose that where , .
We denote by the space of all measurable functions on such that . Equipped with the norm , this is a Banach function space. For the basics on variable exponent Lebesgue spaces, we refer to [2, 29, 30].
We denote by , , the conjugate exponent of . The notation will stand for the set of variable exponents satisfying condition (1) and the local log-condition where does not depend on .
We will use also the following decay conditions:
Let satisfy the log-condition (2). The inequality for bounded open sets is known and proved in . For unbounded sets, if, besides (2), the exponent satisfies the decay condition (3), then we have where See [2, Corollary ]. The following lemma was proved in , Lemma 2, and a simpler proof given in .
We will use a consequence of the estimates of Lemma 1 in the form where we denoted for brevity.
We refer to the appendix for the definition of the classes and used in the following lemma.
Lemma 2. Let and and a function belongs to with respect to uniformly in . Then where does not depend on and .
Proof. Let . We have Since , we obtain By (9), we have Therefore, and we arrive at (11). The last passage to the integral is verified in the standard way with the use of the monotonicity properties of the function in , imposed by the assumptions of the lemma on as
Corollary 3. Let belong to the class uniformly in , where and . Then
Proof. The statement follows from (11) by the definition of the class .
Corollary 4. Let and and a bounded function satisfies the conditions Then where does not depend on and .
Proof. The statement follows directly from (11) with , since when and when .
3. Variable Exponent Morrey Spaces
3.1. Definitions and Some Auxiliary Results for Variable Exponent Morrey Spaces
Let be a nonnegative function on , positive on . Morrey type spaces, called also generalized Morrey spaces, with constant are known in two versions: global and local (we refer, for instance, to the survey paper ) and are defined as the spaces of functions such that respectively, where .
Morrey spaces with variable exponent corresponding to the classical case , but with variable as well, were introduced and studied in . More general approach admitting the variable function were studied in [24, 25].
Definition 5. Let and let be a non-negative function almost increasing in uniformly in . The generalized variable exponent Morrey space is defined by the norm
We will also refer to the space as global generalized variable exponent Morrey space in contrast to its local version defined by the norm where .
For a weight function on , the weighted Morrey space is defined by .
By the definition of the norm in the variable exponent Lebesgue space, we the can also write that From which one can see that for bounded exponents one has
The following lemma provides some minimal assumptions on the function under which the so-defined spaces contain “nice” functions.
Lemma 6. Let . Under the decay condition the assumption is sufficient for bounded functions with compact support (in the case of unbounded set ) to belong to the local Morrey space . Similarly under the log-condition (2), the condition guarantees that such functions belong to the global Morrey space .
We need the following lemma on variable exponent powers of functions in Bary-Stechkin class. For this class, Matuszewska-Orlicz indices, and all the related notation, we refer to the appendix.
Lemma 7. Let , , and satisfy the decay condition (3). Then where and do not depend on and .
Proof. We have to prove that ; that is, It suffices to consider the case . By Theorem A.4, the assumption implies that the function has finite indices and , and holds. Bounds in yield the inequality with some positive and . Then (29) follows from the decay condition at the origin, since .
In papers [17, 18], there were given various conditions for radial type functions to belong to Morrey spaces with nonvariable characteristics. The reader can easily adjust them for the case when they are variable. We do not dwell on this, but in the next lemma we give a certain example of a function in the space , important for our further goals.
Lemma 8. Let and satisfy the decay condition (3) at the origin and for some , and
for small and some . Then the function , where , belongs to .
If additionally we suppose that satisfies the decay condition (4) at infinity, for some , the inequality (30) holds also for large for large and ; then the same holds with .
Proof. We have to check that
By Lemma 7, this is guaranteed by the condition
The latter is equivalent to the condition , that is, , which in its turn is equivalent to (30) and consequently holds.
In the case of , the proof follows the same lines. This time instead of (31) it suffices only to check that for some large . Here by the decay condition at infinity imposed on . Therefore, after which the arguments are similar to those for the case .
3.2. Some Weighted Estimates of Functions in Morrey Spaces
Theorem 10. Let and . Suppose also that and . Then with where and do not depend on and .
Proof. We have
where . Making use of the fact that there exists a such that is almost decreasing, we observe that on . Applying this in (38) and making use of the Hölder inequality with the exponent , we obtain
By (9), we have , so that
It remains to prove that
We have . Since the function is increasing for all and the function is almost decreasing with some , we obtain which proves (41) and completes the proof of the first inequality in (36).
For the second inequality in (36), we proceed in a similar way as where . Since there exists a such that is almost increasing, we obtain Applying the Hölder inequality with the variable exponent and taking (9) into account, we get It remains to prove that , which easily follows by the monotonicity of the involved functions as
4. On Weighted Hardy Operators in Generalized Morrey Spaces
4.1. Pointwise Estimations
We consider the following generalized Hardy operators: where is a non-negative measurable function on . In the one-dimensional case, their versions on the half-axis may be also admitted, so that the sequel with may be read either as or .
We also use the notation .
Our next result on the boundedness of weighted Hardy operators presented in Theorem 13 is prepared by our estimations in Theorem 10. It is in fact a consequence of Theorem 10. We find it useful to divide this consequence into two parts. First, in Theorem 11, we reformulate Theorem 10 in the form to emphasize that we have pointwise estimates of Hardy operators and in terms of the Morrey norm of the function . Then as an immediate consequence of Theorem 11 we formulate Theorem 13 for global Morrey spaces.
Theorem 11. Let and . Let also the weight satisfy the conditions in the case of the operator , and the conditions in the case of the operator . The conditions with , are sufficient for the Hardy operators and , respectively, to be defined on the space . Under these conditions, where
4.2. On the Necessity of the Conditions in (51)
Observe that the conditions in (51) are natural in the sense that they are necessary under some additional assumptions on the function defining the Morrey space.
4.3. Weighted Norm Estimates for Hardy Operators
The statements of Theorem 13 are well known in the case of Lebesgue space, that is, in the case , with constant exponents, when ; see for instance [27, pages 6, 54]. For the classical Morrey spaces with constant exponents and , statements of such type for Hardy operators have been obtained in [17, 19].
Note that, in contrast to variable exponent Lebesgue spaces, inequalities for the Hardy operators in Morrey spaces admit the case when in the case of local Morrey spaces and in the case of global Morrey spaces.
We suppose that the condition holds, which ensures that the space is nonempty by Lemma 6.
Theorem 13. Let , and as well as the functions and satisfy the assumption (54). Let also the weight satisfy the conditions in (49) in the case of the operator and the conditions in (50) in the case of the operator . Then the operators and are bounded from to , if respectively. If and satisfy the assumptions of Corollary 9, then the conditions in (55) are also necessary for the boundedness of the operators and .
Proof. The sufficiency of the conditions in (55) for the boundedness follows from the estimates in (52).
As regards the necessity, the requirements in (55) are nothing else but the statement that respectively, where
The function belongs to by Corollary 9. Consequently, the conditions, (55) are necessary.
Corollary 14. Under the same assumptions on and as in Theorem 13, the Hardy operators and are bounded from the global Morrey space to the global space , if respectively.
Proof. Since , the statement immediately follows from the pointwise estimates in (52).
Remark 15. Theorem 18 is specifically a “Morrey-type” statement in the sense that the case of Lebesgue spaces (the case ) is not included. This, in particular, is reflected in the admission of values in Theorem 18, which is impossible for Lebesgue spaces.
4.4. Finding by a Given
The main theorem of the preceding section, Theorem 13 on the boundedness, provides a relation between the given function and in an indirect form, via the conditions in (55). In the theorems below, under some additional assumptions on the functions we obtain the boundedness conditions in a form of Zygmund-type integral conditions imposed on and give a direct relation between and .
In these theorems, we use the following assumptions on the function defining the data space : where (see for the definition of the classes ). Recall that the assumption in (59) may be equivalently rewritten in terms of the Matuszewska-Orlicz indices of the function as
Proof. By (69), we have To check that , by the definition in (20), we have to estimate the norm We apply Lemma 2 with , which is possible by (59) and obtain Then by (59) we get Therefore, and we arrive at (63).
We single out an important case of non-weighted Hardy operators in variable exponent Morrey spaces of classical type, that is, with the function , defined by where and then by (63).
Proof. In the sufficiency part, the theorem may be