Riemann-Liouville and Higher Dimensional Hardy Operators for NonNegative Decreasing Function in Spaces
One-weight inequalities with general weights for Riemann-Liouville transform and -dimensional fractional integral operator in variable exponent Lebesgue spaces defined on are investigated. In particular, we derive necessary and sufficient conditions governing one-weight inequalities for these operators on the cone of nonnegative decreasing functions in spaces.
We derive necessary and sufficient conditions governing the one-weight inequality for the Riemann-Liouville operator and -dimensional fractional integral operator on the cone of nonnegative decreasing function in spaces.
In the last two decades a considerable interest of researchers was attracted to the investigation of the mapping properties of integral operators in so-called Nakano spaces (see, e.g., the monographs [1, 2] and references therein). Mathematical problems related to these spaces arise in applications to mechanics of the continuum medium. For example, Ružicka  studied the problems in the so-called rheological and electrorheological fluids, which lead to spaces with variable exponent.
Weighted estimates for the Hardy transform in spaces were derived in the papers  for power-type weights and in [5–9] for general weights. The Hardy inequality for nonnegative decreasing functions was studied in [10, 11]. Furthermore Hardy type inequality was studied in [12, 13] by Rafeiro and Samko in Lebesgue spaces with variable exponent.
Historically, one and two weight Hardy inequalities on the cone of nonnegative decreasing functions defined on in the classical Lebesgue spaces were characterized by Arino and Muckenhoupt  and Sawyer , respectively.
It should be emphasized that the operator is the weighted truncated potential. The trace inequity for this operator in the classical Lebesgue spaces was established by Sawyer  (see also the monograph , Ch.6 for related topics).
In general, the modular inequality for the Hardy operator is not valid (see , Corollary 2.3, for details). Namely, the following fact holds: if there exists a positive constant such that inequality is true for all , where ; ; ; and are nonnegative measurable functions, then there exists such that for almost every ; for almost every , and and take the same constant values a.e. for and .
To get the main result we use the following pointwise inequalities: for nonnegative decreasing functions, where , , , and are constants and are independent of , , and , and
In the sequel by the symbol we mean that there are positive constants and such that . Constants in inequalities will be mainly denoted by or ; the symbol means the interval .
We say that a radial function is decreasing if there is a decreasing function such that , . We will denote again by . Let be a measurable function, satisfying the conditions , .
Given such that and a nonnegative measurable function (weight) in , let us define the following local oscillation of : where is the ball with center 0 and radius .
We observe that is nondecreasing and positive function such that where and denote the essential infimum and supremum of on the support of , respectively.
By the similar manner (see ) the function is defined for an exponent and weight on :
Let be the class of nonnegative decreasing functions on and let be the class of all nonnegative radially decreasing functions on . Suppose that is measurable a.e. positive function (weight) on . We denote by the class of all nonnegative functions on for which
Under the symbol we mean the class of nonnegative decreasing functions on from .
Now we list the well-known results regarding one-weight inequality for the operator . For the following statement we refer to .
Theorem A. Let be constant such that . Then the inequity for a weight holds, if and only if there exists a positive constant such that for all
Theorem B (see ). Let be a weight on and such that , and assume that . The following facts are equivalent:(a)there exists a positive constant such that, for any , (b)for any , (c) a.e. and .
Proposition 1. For the operators , and , the following relations hold:(a)(b)
Proof. (a) Upper estimate: represent as follows:
Observe that if , then . Hence
where the positive constant does not depend on and . Using the fact that is decreasing we find that
Lower estimate follows immediately by using the fact that is nonnegative and the obvious estimate and .
(b) Upper estimate: let us represent the operator as follows: Since for we have that Taking into account the fact that is radially decreasing on we find that there is a decreasing function such that Let . Then we have It is easy to see that while using the fact that we find that Finally we conclude that Lower estimate follows immediately by using the fact that is nonnegative and the obvious estimate , where .
We will also need the following statement.
Lemma 2. Let be a constant such that . Then the inequality holds, if and only if there exists a positive constant C such that, for all ,
Proof. We will see that inequality (26) is equivalent to the inequality
where , , and .
Indeed, using polar coordinates in we have
Conversely taking the test function , , in modular inequality (26), one can easily obtain inequality (27).
3. The Main Results
To formulate the main results we need to prove the following proposition.
Proposition 3. Let be a weight on and such that , and assume that . The following statements are equivalent:(a)there exists a positive constant such that, for any , (b)for any , (c) a.e. and .
Proof. We use the arguments of . To show that (a) implies (b) it is enough to test the modular inequality (30) for the function , . Indeed, it can be checked that
Further, we find that Therefore To obtain (c) from (b) we are going to prove that condition (b) implies that is a constant function; namely, for all . This fact and the hypothesis on imply that , and hence, due to (7), Finally (31) means that . Let us suppose that is not constant. Then one of the following conditions holds:(i)there exists such that and, hence, there exists such that or(ii)there exists such that and then, for some , In case (i) we observe that condition (b), for , implies that Then using (36) we obtain, for , which is clearly a contradiction if we let . Similarly in case (ii) let us consider the same condition (b), for , and fix now . Taking into account (38) we find that which is a contradiction if we let .
Finally, the fact that condition (c) implies (a) follows from [18,Theorem 1.7].
Theorem 4. Let be a weight on and such that . Assume that . The following facts are equivalent:(i)there exists a positive constant such that, for any , (ii)condition (13) holds;(iii)condition of Theorem B is satisfied.
Proof. Proof follows by using Theorem B and Proposition 1(a).
Theorem 5. Let be a weight on and such that , and assume that . The following facts are equivalent:(i)there exists a positive constant such that, for any , (ii)condition (31) holds;(iii)condition (c) of Proposition 3 holds.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to Professor A. Meskhi for drawing their attention to the problem studied in this paper and helpful remarks. The authors are also grateful to the editor and anonymous reviewer for their careful review, valuable comments, and remarks to improve this paper.
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