Generalized Fractional Integral Operators on Generalized Local Morrey Spaces

from one generalized local Morrey space LM0 p,φ1 to another LM0 q,φ2 , 1 < p < q < ∞, and from LM{x0} 1,φ1 to the weak spaceWLM0 q,φ2 , 1 < q < ∞. We also find conditions on the pair (φ, ρ) which ensure the Adams-type boundedness of I ρ from M p,φ to M q,φ for 1 < p < q < ∞ and from M 1,φ to WM q,φ for 1 < q < ∞. In all cases the conditions for the boundedness of I ρ are given in terms of Zygmund-type integral inequalities on (φ 1 , φ 2 , ρ) and (φ, ρ), which do not assume any assumption on monotonicity of φ 1 (x, r),


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 secondorder 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  ∈ R  and  > 0, we denote by (, ) the open ball centered at  of radius  and by ∁ (, ) denote its complement.Let |(, )| be the Lebesgue measure of the ball (, ).
By  ≲  we mean that  ≤  with some positive constant  independent of appropriate quantities.If  ≲  and  ≲ , we write  ≈  and say that  and  are equivalent.

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 R  × (0, ∞) and 1 ≤  < ∞.We denote by  , ≡  , (R  ) the generalized Morrey space, the space of all functions  ∈  loc  (R  ) with finite quasinorm: Also by  , ≡  , (R  ) we denote the weak generalized Morrey space of all functions  ∈  loc  (R  ) for which          , = sup According to this definition, we recover the Morrey space  , and weak Morrey space  , under the choice (, ) =  (−)/ :  , =  ,     (,)= According to this definition, we recover the local Morrey space 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 1/+1/  = 1.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 by where  0 is the characteristic function of the unit ball { ∈ R  : || ≤ 1},   is the characteristic function of the annulus . .. By duality, the space   (R  ), called Beurling algebra now, can be described by Let Ḃ  (R  ) and Ȧ (R  ) be the homogeneous versions of   (R  ) and   (R  ) by taking  ∈ Z in (42) and ( 13) instead of  ≥ 0 there.
If  < 0 or  > , then In order to study the relationship between central  spaces and Morrey spaces, Alvárez et al. [19] introduced central bounded mean oscillation spaces and central Morrey spaces The classical result by Hardy-Littlewood-Sobolev states that if 1 <  <  < ∞, then the operator   is bounded from   (R  ) to   (R  ) if and only if  = (1/ − 1/) and for  = 1 <  < ∞, the operator   is bounded from  1 (R  ) to   (R  ) if and only if  = (1 − 1/).Spanne and Adams studied boundedness of the Riesz potential in Morrey spaces.Their results can be summarized as follows.
In [23] the following condition was imposed on (, ): whenever  ≤  ≤ 2, where  ≥ 1 does not depend on ,  and  ∈ R  , jointly with the condition where  > 0 does not depend on  and  ∈ R  .
where  does not depend on  0 and .Then the operator   is bounded from From Theorem 7 we get the following Spanne-type result for   on  , .
where  does not depend on  and .Then the operator   is bounded from  , 1 to  , 2 for  > 1 and from  1, 1 to  , 2 for  = 1.
The following Spanne-type result for   on  , , containing results obtained in [12], was proved in [27].
Then the identity operator  is bounded from Proof.If ,  are nonnegative functions on (0, ∞) and  is nondecreasing, then ess sup Also if ,  are nonnegative functions on (0, ∞) and  is nonincreasing, then ess sup Therefore for all  ∈ A ess sup where First we prove sufficiency.Assume that condition (22) holds.Then for all by (25).
To prove necessity assume that  is bounded from where  > 0 is independent of .
We will use the following statement on the boundedness of the weighted Hardy operator: where  is a weight.
Theorem 11.Let V 1 , V 2 , and  be weights on (0, ∞) and let V 1 () be bounded outside a neighborhood of the origin.The inequality holds for some  > 0 for all nonnegative and nondecreasing  on (0, ∞) if and only if Moreover, the value  =  is the best constant for (33).

Spanne-Type Result for the Operator 𝐼
We assume that so that the fractional integrals    are well defined, at least for characteristic functions 1/|| 2 of complementary balls: In addition, we will also assume that  satisfies the growth condition: there exist constants  1 > 0 and 0 < 2 This condition is weaker than the usual doubling condition for the function ()/  : there exists a constant  2 > 0 such that whenever  and  satisfy ,  > 0 and 1/2 ≤ / ≤ 2.
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 G0 .We will write, when  ∈ G0 , Remark 13.Typical examples of () that we envisage are, for 0 <  < , and, for  > 0, The second one is used to control the Bessel potential (see also [30]).
The following theorem is one of the main results of this paper.

Adams-Type Result for
the Operator   in  , The following Adams-type result was proved in [31] (see also [12]).
where  does not depend on  ∈ R  and  > 0.
The following Theorem was proved in [32].
where  does not depend on  and .Then, for  > 1, the Hardy-Littlewood maximal operator  is bounded from  , 1 to  , 2 and, for  = 1,  is bounded from  1, 1 to  1, 2 .
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 1 ≤  < ∞,  > , and () satisfy the conditions (37) and (42).Let also (, ) satisfy the condition (60) and where  does not depend on  ∈ R  and  > 0.