Weighted Hardy-Type Inequalities in Variable Exponent Morrey-Type Spaces

and Applied Analysis Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 ISRN Applied Mathematics Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 International Journal of Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 Journal of Function Spaces and Applications International Journal of Mathematics and Mathematical Sciences Hindawi Publishing Corporation http://www.hindawi.com Volume 2013


Introduction
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  (⋅) (Ω), Ω ⊆ R  , known also as Lebesgue spaces with variable exponent ().We refer to the existing books [1][2][3] in the field.
This variable exponent boom naturally touched Morrey spaces.Morrey spaces L , (with constant exponents) in its classical version were introduced in [4] in relation to the study of partial differential equations and presented in various books; see, for example, [5][6][7]; we refer also to a recent overview of Morrey spaces in [8], where various generalizations of Morrey spaces may be also found.
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 Morreytype 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 (0, ) we obtain the boundedness conditions in a different form via Zygmund-type conditions on (0, )/() and provide a direct relation between (0, ) and (0, ).
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 (0) 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.
We will use also the following decay conditions: For brevity, by P 0,∞ (Ω) we denote the set of bounded measurable functions (not necessarily with values in [1, ∞)), which satisfy the decay conditions (3) and (4).
We will use a consequence of the estimates of Lemma 1 in the form      (0,2)\(0,) where we denoted for brevity.We refer to the appendix for the definition of the classes (R + ) and (R + ) used in the following lemma. / * ()  (, )   ,  > 0, (11) where  > 0 does not depend on  and .
Proof.Let  (0,) := (0, 2 − ) \ (0, 2 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 The class Z  0 , ∞ (R 1 + ) used in the following corollary is defined in (A.18).
Proof.The statement follows directly from (11) with (, ) =  ]() , since (we refer, for instance, to the survey paper [8]) and are defined as the spaces of functions  ∈

Definitions and Some Auxiliary Results for Variable
respectively, where  0 ∈ Ω.
Morrey spaces with variable exponent () corresponding to the classical case () =  / , but with variable () as well, were introduced and studied in [21].More general approach admitting the variable function (, ) were studied in [24,25].
Following [24], we introduce the variable exponent Morrey-type space by the definition below, but note that our notation differs from that of [24].Definition 5. Let 1 ≤  − ≤  + < ∞ and let (, ) be a nonnegative function almost increasing in  uniformly in  ∈ Ω.The generalized variable exponent Morrey space L (⋅),(⋅) (Ω) is defined by the norm We will also refer to the space L (⋅),(⋅) (Ω) as global generalized variable exponent Morrey space in contrast to its local version L (⋅),(⋅)  0 ;loc (Ω) defined by the norm where  0 ∈ Ω.
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. (, ) < ∞ (27) guarantees that such functions belong to the global Morrey space L (⋅),(⋅) (Ω).
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.
and  satisfy the decay condition (3).Then where ℓ < ∞ and  ≥ 1 do not depend on  and .

Proof
By Lemma 7, this is guaranteed by the condition The latter is equivalent to the condition ( (0) (0, )) > 0, that is, ((0, )) > 0, 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) after which the arguments are similar to those for the case  < ∞.
Proof.We have where   () = { : 2 By ( 9), we have It remains to prove that We have Since the function  /  * () is increasing for all  ∈ R + and the function (0, )/  () 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   () = { : 2   < || < 2 +1 }.Since there exists a  ∈ R such that   V() is almost increasing, we obtain Applying the Hölder inequality with the variable exponent (⋅) and taking ( 9) into account, we get /  * (2  ) (2  ) ≤ B(), which easily follows by the monotonicity of the involved functions as

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 R  .In the one-dimensional case, their versions on the half-axis R 1 + may be also admitted, so that the sequel R  with  = 1 may be read either as R 1 or R 1 + .We also use the notation  (⋅) =  (⋅)  | ≡1 .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 H (⋅)  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.

On the
Hardy inequalities in the variable exponent Lebesgue spaces were studied in [34][35][36]; see also the references therein.
Note that, in contrast to variable exponent Lebesgue spaces, inequalities for the Hardy operators in Morrey spaces admit the case inf  () = 1 when (0, 0) = 0 in the case of local Morrey spaces and sup  (, 0) = 0 in the case of global Morrey spaces.
Theorem 13.Let 1 ≤  − ≤  + < ∞, 1 ≤  − ≤  + < ∞ and  ∈ P 0,∞ (Ω) 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 H (⋅)  .Then the operators  (⋅)  and respectively.If  and (0, ) satisfy the assumptions of Corollary 9, then the conditions in (55) are also necessary for the boundedness of the operators  (⋅)  and H (⋅)  .
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  0 belongs to L (⋅),(⋅) 0,loc (R  ) 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 H (⋅)  are bounded from the global Morrey space L (⋅),(⋅) (R  ) to the global space Proof.Since ‖‖ L (⋅),(⋅) 0,loc (R  ) ≤ ‖‖ L (⋅),(⋅) , 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  = 0) is not included.This, in particular, is reflected in the admission of values () = 1 in Theorem 18, which is impossible for Lebesgue spaces.
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.Let  be as in Theorem 11 and (0, ⋅) ∈ ([0,  0 ]) ∩ ([0,  0 ]) for some  0 > 0, and (30) holds.Then the conditions in (51) are necessary for the Hardy operators  (⋅)  and H (⋅)  , respectively, to be defined on the space L 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  ≡ 1, with constant exponents, when 1 <  < /; see for instance[27, pages 6, 54].For the classical Morrey spaces L , (R  ) with constant exponents  and , statements of such type for Hardy operators have been obtained in