Boundedness of Riesz Potential Operator on Grand Herz-Morrey Spaces

: In this paper, we introduce grand Herz–Morrey spaces with variable exponent and prove the boundedness of Riesz potential operators in these spaces.


Introduction
In the last two decades, under the influence of some applications revealed in [1], there was a vast boom of research of the so-called variable exponent spaces, and the operator in them. For the time being, the theory of such variable exponent Lebesgue, Orlicz, Lorentz and Sobolev function spaces is widely developed, and we refer to the books [2,3] and the surveying papers [4][5][6][7]. Herz spaces with variable exponents have been recently introduced in [8][9][10]. In [11], variable parameters were used to define continual Herz spaces, and proved the boundedness of sublinear operators in these spaces. The boundedness of other operators such as Riesz potential operator and the Marcinkiewicz integrals was proved in [12,13].
The concept of Morrey spaces L p,λ was introduced by C. Morrey in 1938 (see [14]) in order to study regularity questions that appear in the calculus of variations. They describe local regularity more precisely than Lebesgue spaces and are widely used not just in harmonic analysis, but also in PDEs. Meskhi introduced the idea of grand Morrey spaces L r),θ,λ and derived the boundedness of a class of integral operators (Hardy-Littlewood maximal functions, Calderón-Zygmund singular integrals and potentials) in these spaces, see ( [15]). Moreover, Izuki [16] defined the Herz-Morrey spaces with a variable exponent and investigated the boundedness of fractional integrals on these spaces.
In [17], the idea of grand variable Herz spacesK α,p),θ q(·) (R n ) was introduced and proved the boundedness of sublinear operatorsK α,p),θ q(·) (R n ). Motivated by the concept, in this article, we introduce the concept of grand Herz-Morrey spaces, and prove the boundedness of the Riesz potential operator on grand Herz-Morrey spaces with variable exponents. There are four sections in this article; the first section is dedicated to the introduction, the second section contains some basic definitions and lemmas, we introduce the concept of grand Herz-Morrey spaces in part three, and the boundedness of the Riesz potential operator on grand variable Herz-Morrey spaces is proved in the last section.

Lebesgue Space with Variable Exponent
Assume that G ⊆ R n is an open set and p(·) : G → [1, ∞) is a real-valued measurable function. Let the following condition hold: where (i) p − := ess inf g∈G p(g) (ii) p + := ess sup g∈G p(g).
Lebesgue space L p(·) (G) is the space of measurable functions f 1 on G such that, norm is defined as, this is the Banach function space, p (g) = p(g) p(g)−1 denotes the conjugate exponent of p(g). Next, we will define the space L p(·) loc (G) as, L p(·) loc (G) := κ : κ ∈ L p(·) (K) for all compact subsets K ⊂ G . Now to define the log-condition, where C = C(η) > 0 is not dependent on z 1 , z 2 . For the decay condition: let η ∞ ∈ (1, ∞), such that inequality (4) holds for η 0 ∈ (1, ∞) in case of homogenous Herz spaces. We adopted the following notations in this paper: where D(g, t) := {y ∈ G : |g − y| < t}. (ii) The set P (G) is the collection of all p(·) satisfying p − > 1 and p + < ∞. (iii) P log = P log (G) is the class of functions p ∈ P (G) satisfying (1) and (2). (iv) When G is unbounded, P ∞ (G) and P 0,∞ (G) are the subsets of P (G) and its values lies in [1, ∞) satisfying (3) and (4), respectively.
(v) In the case G is bounded, P ∞ (G) and P 0,∞ (G) are the subsets of P (G).
(vi) In the case S is unbounded, P ∞ (S) are the subsets of exponents in L ∞ (S) and its values lies in [1, ∞], which satisfy both conditions (2) and (3), respectively, and P log ∞ (S) is the set of exponents p ∈ P ∞ (S), satisfying condition (1). C is a constant that is independent of the main parameters involved, and its value varies from line to line. Lemma 1 ([11]). Let D > 1 and η ∈ P 0,∞ (R n ). Then, respectively, where k 0 ≥ 1 and k ∞ ≥ 1 depend on D and do not depend on t.

Herz Spaces with Variable Exponent
We adopted the following notations in this subsection: Definition 1. Let r, s ∈ [1, ∞), α ∈ R, the classical versions of Herz spaces, commonly known as non-homogenous and homogenous Herz spaces, can be defined by the norms, g Kα r,s (R n ) := respectively, where R t,τ stands for the annulus R t,τ := D(0, τ)\D(0, t).

Herz-Morrey Spaces
Next, we define Herz-Morrey spaces with variable exponent.

Grand Lebesgue Sequence Space
Now, we will define the grand Lebesgue sequence space. G is representing one of the sets N 0 , Z n , N, Z in the following definitions (see [19]).

Definition 5.
Let r ∈ [1, ∞) and θ > 0. The grand Lebesgue sequence space l r)θ can be defined by the norm The following nesting properties hold:

Grand Variable Herz-Morrey Spaces
Grand variable Herz-Morrey spaces are introduced in this section.
We define the homogeneous grand variable Herz-Morrey spaces can be defined by the norm: . For λ = 0, the grand Herz-Morrey spaces become grand Herz spaces.
Non-homogeneous grand variable Herz-Morrey spaces can be defined in a similar way. .
Proof. We have , by using Hölder's inequality , by using generalized Hölder's inequality

Boundedness of the Riesz Potential Operator
Now Riesz potential operator can be defined as with the normalizing constant η n (γ) = 2 γ π n 2 Γ(γ/2) Γ((n−γ)/2) . Whenever γq 1 (z 1 ) < n, we can define the Sobolev conjugate of q 1 by the usual relation The well-known Sobolev theorem was extended to variable exponents in [20] for bounded sets in R n under the assumption that the maximal operator is bounded in L p(·) (Ω); for unbounded sets, proved in [21], the Sobolev theorem runs as follows.
For E 22 , we use the fact 2 kα(z 1 ) = 2 kα ∞ , k ≥ 0, z 1 ∈ R k , we obtain For each k ∈ Z and l ≤ k − 2 and a.e. z 1 ∈ R k , z 2 ∈ R l , we have splitting E 1 by using Minkowski's inequality we have By using Lemma 1, we have applying above estimates to E 11 , we can obtain by using Hölder's inequality, Fubini's theorem and the inequality 2 −r(1+δ) < 2 −r , we obtain . Now, for E 12 using Minkowski's inequality, we have The estimate for A 2 can be obtained by similar way to E 11 by replacing q 1 (0) with q 1 ∞ and using the fact n q 1∞ − α ∞ > 0. For A 1 using Lemma 1, we obtain . Now, by using Hölder's inequality and the fact n q 1 (0) − α(0) > 0, we have . Now, we estimate E 3 , for every k ∈ Z and l ≥ k + 2 and a.e. z 1 ∈ R k ; the size condition and Hölder's inequality imply splitting E 3 by applying the Minkowski's inequality we have For E 32 Lemma 1 yields we get where d = n q 1∞ + α ∞ > 0. Then, we use Hölder's theorem for series and 2 −r(1+δ) < 2 −r to obtain . Now, for E 31 using Monkowski's inequality, we have The estimate for B 1 can be obtained similar to E 32 by replacing q 1 ∞ with q 1 (0) and applying the fact that n q 1 (0) + α(0) > 0. For B 2 using Lemma 1, we obtain . Now, by using Hölder's inequality and the fact that n q ∞ + α ∞ > 0, we have Combining the estimates for E 1 , E 2 and E 3 yields I γ f MK α(·),r),θ λ,q 2 (·) (R n ) ≤ C f MK α(·),r),θ λ,q 1 (·) (R n )

Conclusions
We have defined a new type of space called variable exponents grand Herz-Morrey spaces, where we used discrete grand spaces, and we have proved the boundedness of the Riesz potential operator on these spaces.