Riemann-Liouville and Higher Dimensional Hardy Operators for NonNegative Decreasing Function in L p ( ⋅ ) Spaces

and Applied Analysis 3 Observe that if t < x/2, then x/2 < x − t. Hence S 1 (x) ≤ c 1 x ∫ x/2 0 f (t) dt ≤ cTf (x) , (17) where the positive constant c does not depend on f and x. Using the fact that f is decreasing we find that S 2 (x) ≤ cf ( x 2 ) ≤ cTf (x) . (18) Lower estimate follows immediately by using the fact that f is nonnegative and the obvious estimate x − t ≤ x and 0 < t < x. (b) Upper estimate: let us represent the operator I α as follows: I α g (x) = 1 |x| α ∫ |y|<|x|/2 g (y) 󵄨󵄨󵄨󵄨x − y 󵄨󵄨󵄨󵄨 n−α dy


Introduction
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 [3] studied the problems in the so-called rheological and electrorheological fluids, which lead to spaces with variable exponent.
Historically, one and two weight Hardy inequalities on the cone of nonnegative decreasing functions defined on R + in the classical Lebesgue spaces were characterized by Arino and Muckenhoupt [18] and Sawyer [19], 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 [20] (see also the monograph [21], Ch.6 for related topics).
In general, the modular inequality for the Hardy operator is not valid (see [22], Corollary 2.3, for details).Namely, the following fact holds: if there exists a positive constant  such that inequality ( * ) is true for all  ≥ 0, where ; ; ; and V are nonnegative measurable functions, then there exists  ∈ [0, 1] such that () > 0 for almost every  < ; V() = 0 for almost every  > , and () and () take the same constant values a.e. for  ∈ (0; ) and  ∈ (0; ) ∩ {V ̸ = 0}.To get the main result we use the following pointwise inequalities: for nonnegative decreasing functions, where  1 ,  2 ,  3 , and  4 are constants and are independent of , , and , and In the sequel by the symbol  ≈  we mean that there are positive constants  1 and  2 such that  1 () ≤ () ≤  2 ().Constants in inequalities will be mainly denoted by  or ; the symbol R + means the interval (0, +∞).

Preliminaries
We say that a radial function  : R  → R + is decreasing if there is a decreasing function  : R + → R + such that (||) = (),  ∈ R  .We will denote  again by .Let  : R  → R  be a measurable function, satisfying the conditions Given  : R  → R + such that 0 <  − ≤  + < ∞ and a nonnegative measurable function (weight)  in R  , let us define the following local oscillation of : where (0, ) is the ball with center 0 and radius .
We observe that  (⋅), () is nondecreasing and positive function such that lim where  +  and  −  denote the essential infimum and supremum of  on the support of , respectively.
By the similar manner (see [10]) the function  (⋅), () is defined for an exponent  : R +  → R + and weight V on R + : Let (R + ) be the class of nonnegative decreasing functions on R + and let (R  ) be the class of all nonnegative radially decreasing functions on R  .Suppose that  is measurable a.e.positive function (weight) on R  .We denote by  () (, R  ) the class of all nonnegative functions on R  for which For essential properties of  () spaces we refer to the papers [23,24] and the monographs [1,2].
Under the symbol  (x) dec (, R + ) we mean the class of nonnegative decreasing functions on R + from  () (, R  ) ∩ (R  ).
Now we list the well-known results regarding one-weight inequality for the operator .For the following statement we refer to [18].
Theorem A. Let  be constant such that 0 <  < ∞.Then the inequity for a weight V holds, if and only if there exists a positive constant  such that for all  > 0 Condition ( 11) is called   condition and was introduced in [18].
Proposition 1.For the operators , ,   , and   , the following relations hold: (a) Proof.(a) Upper estimate: represent    as follows: Observe that if  < /2, then /2 <  − .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 0 <  < .
(b) Upper estimate: let us represent the operator   as follows: Since ||/2 ≤ | − | for || < ||/2 we have that Taking into account the fact that  is radially decreasing on R  we find that there is a decreasing function  : R + → R + such that It is easy to see that while using the fact that /( − ) > 1 we find that Finally we conclude that Lower estimate follows immediately by using the fact that  is nonnegative and the obvious estimate | − | ≤ ||, where 0 < || < ||.
We will also need the following statement.

The Main Results
To formulate the main results we need to prove the following proposition.
In case (i) we observe that condition (b), for  = , implies that Proof.Proof follows by using Propositions 3 and 1(b).