Boundedness of commutators of variable Marcinkiewicz fractional integral operator in grand variable Herz spaces

Let S n –1 denote unit sphere in R n equipped with the normalized Lebesgue measure. Let (cid:2) ∈ L s ( S n –1 ) be a homogeneous function of degree zero such that (cid:2) S


Introduction
It is well known that Beurling [8] and Herz [16] introduced new spaces to characterize certain properties of functions.These spaces are known as Herz spaces, and numerous studies involving them can be found in the literature.One of the main reasons is that the Hardy space theory associated with Herz spaces is very rich.These new Hardy spaces serve as a localized version of the ordinary Hardy spaces and can sometimes be better substitutes when considering, for instance, the boundedness of non-translation invariant singular integral operators.
Nowadays, there is a vast boom of research related to both the study of the Herz spaces themselves and the operator theory in these spaces.This is caused by the influence of some possible applications in modeling with nonstandard local growth (in differential equations, fluid mechanics, elasticity theory; see, for example, [1-4, 7, 23, 25, 26]).There was a vast boom of research in the so-called variable exponent spaces.We refer, for example, to the papers [9,[11][12][13][14], see also references therein.
In [20], authors considered variable potential operators I β(x) to prove a Sobolev-type theorem for the potential operator from the Lebesgue space L p(•) into the weighted Lebesgue space L q(•) w in R n , under the conditions that p(x) is satisfying the logarithmic condition locally and at infinity.It was not supposed that p(x) is constant at infinity but also assumed that p(x) took its minimal value at infinity.
Motivated by the above results, in this paper, we prove the boundedness of the commutators of variable Marcinkiewicz fractional integral operator on grand variable Herz spaces Kα(•),q),θ

Preliminaries
If H is a measurable set in R n and q(•): where q -:= ess inf h∈H q(h), q + := ess sup h∈H q(h).
(a) Lebesgue space with variable exponent L q(•) (H) is defined as dy < ∞, where γ is a constant .
Norm in L q(•) (H) is defined as loc (H) can be defined as Let us recall the well-known log-Hölder continuity condition (or the Dini-Lipschitz condition) for q : H → (0, ∞): there is a positive constant C such that for all x, y ∈ H with Further, we say that q(•) satisfies the decay condition if there exists and there is a positive constant C ∞ > 0 such that We will also need the log-Hölder continuity condition at 0 for q(•): there are constants C 0 > 0 such that for all |h| ≤ 1 2 , The best possible constant C in (2.2) (resp.C ∞ in (2.3)) is called log-Hölder continuity or log-Dini-Lipschitz constant (resp.decay constant) for the exponent q(•).
We use these notations in this article: (i) B(x, r) is the ball of radius r and center at the point x.
H) be a locally integrable function, then the Hardy-Littlewood maximal operator M is defined as (vii) The set P(H) consists of all measurable functions q(•) satisfying q -> 1 and q + < ∞. (viii) P log = P log (H) consists of all functions q ∈ P(H) satisfying ( In what follows, we denote By p (x) = p(x)/(p(x) -1), we denote the conjugate exponent of p(

•). (xii)
C is a constant, its value varies from line to line and is independent of main parameters involved.
Lemma 2.1 [27] Let D > 1 and q ∈ P 0,∞ (R n ).Then and respectively, where c 0 ≥ 1 and c ∞ ≥ 1 depend on D but do not depend on r.
The Hölder inequality in the variable exponent case has the following form: where κ = 1 p -+ 1 (p ) -, and is a measurable subset of R n .The next statement is the generalized Hölder inequality for variable exponent Lebesgue spaces (see [10]): Lemma 2.2 Let H be a measurable subset of R n and p(•) be an exponent such that Definition 2.3 (BMO space) A BMO function is a locally integrable function u whose mean oscillation given by 1 |B| B |u(y)u B |dy is bounded.Mathematically, The homogeneous Herz space Kα(•) Next we will define grand variable Herz spaces.
Then the grand variable Herz space Kα(•),p),θ The next proposition is the generalization of variable exponents Herz spaces in [6].We omit the proof of proposition 2.6 since it is essentially similar to the proof given in [6], and with slight modification, we can obtain the following result in grand variable Herz spaces.Proposition 2.7 Let α, p, q be as defined in definition 2.5, then . The Riesz-type potential operator of variable order β(x) is defined by (2.7) Note that the β(x) is the order of the Riesz potential operator, which is variable.We are assuming that the order of Riesz potential operator β(x) is not continuous rather, we are assuming that it is a measurable function in R n satisfying the following conditions: ess sup The following proposition is one of the main requirements for proving our main results.The given proposition was proved in [20] and is commonly known as a Sobolev theorem for the Riesz potential operator in Lebesgue spaces.
Let S n-1 denote the unit sphere in R n (n ≥ 2) with the normalized Lebesgue measure.Let ∈ L r (S n-1 ) be a homogeneous function of degree zero such that where y = y/|y|, and y is not zero.The Marcinkiewicz integral was introduced by Stein [28] in connection with Littlewood-Paley g-function on R n as: , where Let b be a locally integrable function on R n .The commutators on the variable Marcinkiewicz fractional integral operator are defined by

The commutators of variable Marcinkiewicz fractional integral operator on grand variable Herz spaces
Let be homogeneous of degree zero and ∈ L s (S n-1 ), s > q -.Let α be such that: , and For E 1 , splitting E 1 using Minkowski's inequality, we have For estimating E 11 , we use the facts that, for each k ∈ Z and l ≤ k and a.e.
By the virtue of the mean value theorem we obtain, (3.1) For I 11 , using Minkowski's inequality, generalized Hölder's inequality, and inequality (3.1), we have Similarly, for I 12 , we have We define q 1 (•) by the relation 1 q 1 (x) = 1 q 1 (x) + 1 s .Using Lemma (2.9) and generalized Hölder's inequality, we have Similarly, using Lemma (2.4), we have It is known [37] that Consequently, by proposition (2.8), we have As a result, we get Applying results to E 11 , we can get 0) > 0, applying Hölder's inequality, 2 -p(1+ψ) < 2 -p and Fubini's theorem for series, we get . Now, for E 12 , using Minkowski's inequality, we have The estimate for A 2 follows in a similar manner to E 11 by replacing q 1 (0) with q 1∞ and by the use Now using the fact that - Now we estimate E 2 , for each k ∈ Z and l ≥ k + 1 and a.e.
Using similar arguments as used in I 11 , we obtain Using the same arguments of I 12 , we obtain ) gχ l L q 1 (•) .
Using these estimates for B 2 , we get the estimate for E 21 .