Variable Herz–Morrey estimates for rough fractional Hausdorff operator

As a ﬁrst attempt, we obtain the boundedness of the rough fractional Hausdorﬀ operator on variable exponent Herz-type spaces. The method used in this paper enables us to study the operator on some other function spaces with variable exponents


Introduction
The Hausdorff operator has a fascinating history of its development from simple to present form.The laying stone of this development is the one-dimensional Hausdorff operator where ∈ L 1 loc (R).A change of variables in (1.1) results in the following form: The boundedness of these operators, an essential part of the analysis, has been discussed in [10,29,30].For a detailed history and the recent development of the Hausdorff operator, we refer the interested reader to the review papers [6,28].The Hausdorff operator finds its applications in the study of one-dimensional Fourier analysis.In particular, it plays a significant role in the summability of the classical Fourier series.Therefore extensions of the one-dimensional Hausdorff operators to multidimensional spaces become extremely important.Several authors have spared their studies for such an extension.In this regard, some contributions are [5,26,27,31].Here we are mainly interested in the rough Hausdorff operator [5] |t| n t g(t) dt (1.3) and the fractional Hausdorff Hausdorff operator [31] |t| n-β g(t) dt, 0≤ β < n. (1.4) The rough fractional Hausdorff operator, which is a combination of (1.3) and (1.4), |t| n-β t g(t) dt, 0≤ β < n, (1.5) was first studied in [14].It is obvious that by taking β = 0 in (1.5) we obtain H , , and by taking = 1 in the same identity, we get the fractional Hausdorff operator H β .Thus discussing the boundedness of H β , on variable exponent Herz-type spaces automatically includes the same results for H , and H β on these spaces.Some studies containing boundedness results regarding H , , H ,1 , and H β along with their commutators on different function spaces include [2, 3, 7, 11-13, 17, 19, 20, 35].
Another important aspect of the Hausdorff operator is that it contains other classical operators as its particular cases.For example, in , respectively, then we obtain the n-dimensional fractional Hardy operator and its adjoint operator: During the recent past, the boundedness of Hardy-type operators on variable exponent function spaces also drew great attention of the research community [18,39].Besides the Hardy operators, the Hausdorff operator also contains the n-dimensional version of the Caldrón operator Variable exponent function spaces have an increasing impact on the recent advances in harmonic analysis.This is mainly because of their frequent appearance in analysis and applications of different functional analysis tools for partial differential equations.This results in an increase in research publications in this field.The theory of variable Lebesgue spaces first appeared in the pioneering work of Kováčik and Rákosník [25].Later on, several monograms appeared in the literature to strengthen the theory.Operator theory on function spaces also finds new dimensions, and many researchers discuss the boundedness of different operators on variable-exponent function spaces [24,37].In this paper, we inquire about the continuity of the rough fractional Hausdorff operator from this perspective.
On function spaces with constant exponents, the boundedness of the Hausdorff operator is accomplished by a scaling argument followed by the polar decomposition of integral on R n .However, such a direct approach on function spaces with variable exponent does not work well.It needs some modification in the later case.In this paper, we aim to tackle this problem by establishing the boundedness of the rough fractional Hausdorff operator on variable-exponent Herz-type spaces.Our strategy enables us to study the operator on other function spaces with variable exponents.As corollaries of our main results, we prove the boundedness of the rough Hausdorff operator on variable-exponent Herz-type spaces, which are also new to the best of the author's knowledge.
In the next section, we present some definitions and preliminary lemmas.Section 3 contains the theorems stating the boundedness criterion for rough fractional Hausdorff operators on variable-exponent Herz-type spaces and their detailed proofs.

Variable-exponent function spaces
Let O ⊆ R n be an open set, and let q(•) : O → [1, ∞) be a measurable function.We denote by q (•) the conjugate exponent of q(•) satisfying The set P(O) consists of all q(•) such that The Lebesgue space with variable exponent L q(•) (O) is a set of all measurable functions g such that for a positive σ , Equipped with the Luxemburg norm dy ≤ 1 , the space L q(•) (O) becomes a Banach function space.The set of functions serves to define local version of the variable-exponent Lebesgue space L q(•) loc (O).Variable-exponent function spaces bear a deep connection with the boundedness of Hardy-Littlewood maximal operator M defined by ) , where C is a positive constant independent of ξ and η. where where r q,q 1 = (1 + 1 (q 1 ) --1 (q 1 ) + ) 1/q -.

Lemma 2.3 ([22]
) If q(•) ∈ B(R n ), then there exist constants 0 < δ < 1 and C > 0 such that for all balls B in R n and all measurable subsets S ⊂ B, , for all balls B ⊂ R n and for a positive constant C, we have The boundedness of the fractional integral operator I β defined by on variable Lebesgue space (see [4]) takes a crucial part in proving our main result.
Lemma 2.8 Let β, q 1 (•), and q 2 (•) be as defined in Proposition 2.7.Then Then the homogenous Herz space with variable exponent is defined as follows.
If p(•) = p, then we have the classical Herz space Kα,p q defined in [32].Some generalizations of Herz spaces were made in [1,15,16,36] shortly after their first appearance. where ) is the Herz space with variable exponent.The Herz-Morrey spaces with variable exponent M Kα,λ q,p(•) are first defined in [21,22].

If is a radial function and A
(3.1) Proof Since q 1 ≤ q 2 , by the definition of the Morrey-Herz space we have For both I 1 and I 2 , we have to approximate the inner norm (H For this, we proceed as follows: Since p 1 (•) ∈ P(R n ), we can fix s such that s > p 1 (•).We define a new variable p(•) such that 1 p 1 (•) = 1 s + 1 p(•) .So by Lemma 2.5 we obtain By polar decomposition it is easy to see that (|x|r -1 ) r n-β s y s dσ y r n dr r , where dσ (y ) denotes the normalized Lebesgue measure on the unit sphere S n-1 .A change of variables results in the following inequality: Also, when |B j | ≤ 2 n and y ∈ B j , by 1 p 1 (y) = 1 s + 1 p(y) and Lemma 2.6 we have When |B j | ≥ 1, we have Hence we obtain Multiplying both sides with χ k (x) and taking the L p 2 (•) norm, the last inequality becomes Having estimating the inner norm, we are now in position to approximate I 1 and I 2 .Let us first approximate I 1 .For j < k, in view of Remark 2.11, from (3.5) we obtain the following inequality: The conditions 0 < β < n (p 1 ) + and 1 p 2 (•) + β n = 1 p 1 (•) ensure the applicability of Lemma 2.8 to obtain Finally, Lemma 2.4 helps us in establishing the following inequality: Hence, using (3.6), we get The condition α < nδ 1 -n (p 1 ) + implies that α < n(δ 1 -1 s ).So for 1 < q 1 < ∞, we use the Hölder inequality to obtain Similarly, for 0 < q 1 ≤ 1, from (3.7) we have .
Next, we turn to estimate I 2 .For j ≥ k, we again use inequality (3.5) and Remark 2.11 to obtain Making use of the Lemma 2.8 once again, we get which in view of Lemma 2.4 results in Hence I 2 can be approximated as Similarly, the condition 0 So we can choose a constant > 1 such that λ -1 (α + β + nδ 2 -n s ) < 0. For 1 < q 1 < ∞, by the Hölder inequality we get In view of the same condition n snδ 2β < α, M 1 is approximated as Since λ < 1 (α + β + nδ 2 -n s ) and α + β + nδ 2 -n s > 0, we get This completes the approximation of I 2 in the case 1 < q 1 < ∞.
Finally, it remains to estimate I 2 for the case 0 < q 1 ≤ 1.For this, from (3.8) we get Since α + β + nδ 2 -n s > 0, for L 1 , we get Lastly, for the estimation of L 2 , we proceed as follows: .
Combining all the estimates we arrive at (3.1).Thus we have completed the proof of the theorem.

Conclusions
In this paper, we have shown that the rough fractional Hausdorff operator is bounded on Herz-type spaces.The scaling argument commonly employed to establish the boundedness of the Hausdorff operator on function spaces with constant exponents makes the study of Hausdorff operators on function spaces with variable exponents unsuitable.We overcome this problem by employing a new strategy.Furthermore, this strategy will be helpful in studying Hausdorf-type operators on other function spaces with variable exponents.