KYUNGPOOK Math. J. 2019; 59(2): 259-275
On the Boundedness of Marcinkiewicz Integrals on Variable Exponent Herz-type Hardy Spaces
Rabah Heraiz
Department of Mathematics, Laboratory of Functional Analysis and Geometry of Spaces, M’sila University, P. O. Box 166, M’sila 28000, Algeria
e-mail : heraizrabeh@yahoo.fr and rabah.heraiz@univ-msila.dz
* Corresponding Author.
Received: March 22, 2018; Revised: November 2, 2018; Accepted: December 6, 2018; Published online: June 23, 2019.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

The aim of this paper is to prove that Marcinkiewicz integral operators are bounded from $K˙p(·)α(·),q(·)(ℝn)$ to $K˙p(·)α(·),q(·)(ℝn)$ when the parameters α(·), p(·) and q(·) satisfies some conditions. Also, we prove the boundedness of μ on variable Herz-type Hardy spaces $HK˙p(·)α(·),q(·)(ℝn)$.

Keywords: Herz spaces, Herz-type Hardy spaces, variable exponent, HardyLittlewood maximal operator, Marcinkiewicz integral operators.
1. Introduction and Preliminaries

Function spaces with variable exponent are being actively studied not only in the field of real analysis but also in partial differential equations and in applied mathematics. The theory of function spaces with variable exponents has rapidly made progress in the last three decades.

For 0 < β ≤ 1, the Lipschitz space Lipβ(ℝn) is defined as

$Lipβ(ℝn):={f:‖f‖Lipβ(ℝn)=supx,y∈ℝn;x≠y∣f(x)-f(y)∣∣x-y∣β<∞}.$

Given Ω ∈ Lipβ(ℝn) be a homogeneous function of degree zero and

$∫Sn-1Ω(x′) dσ(x′)=0$

where x′ = x/|x| for any x ≠ 0 and Sn−1 denotes the unit sphere in ℝn (n ≥ 2) equipped with the normalized Lebesgue measure.

The Marcinkiewicz integral μ is defined by

$μ(f) (x):=(∫0∞∣FΩf(x)∣2dtt3)12$

where

$FΩf(x):=∫∣x-y∣≤tΩ(x-y)∣x-y∣n-1f(y) dy.$

It is well known that the operator μ was first defined by Stein [13] and under the conditions above, Stein proved that μ is of type (p, p) for 1 < p ≤ 2 and of weak type (1, 1). Benedek et al. [2] showed that μ of type (p, p) with 1 < p < ∞.

Recently, the boundedness of Marcinkiewicz integral operators μ on variable function spaces have attracted great attention (see [14, 15, 18] and their references).

The purpose of this paper is to generalize some results concerning Marcinkiewicz integral operators μ on variable Herz spaces $K˙p(·)α(·),q(·)(ℝn)$ and variable Herz-type Hardy spaces $HK˙p(·)α(·),q(·)(ℝn)$. We define the set of variable exponents by

$P0(ℝn):={p measurable:p(·):ℝn→[c,∞] for some c>0}.$

The subset of variable exponents with range [1,∞) is denoted by 𝒫(ℝn). For p ∈ 𝒫0(ℝn), we use the notation

$p-=essinfx∈ℝnp(x), p+esssupx∈ℝnp(x).$

### Definition 1.1

Let p ∈ ℘0(ℝn). The variable exponent Lebesgue space Lp(·)(ℝn) is the class of all measurable functions f on ℝn such that the modular

$ϱp(·) (f):=∫ℝn∣f(x)∣p(x)dx$

is finite. This space is a quasi-Banach function space equipped with the norm

$‖f‖p(·):=inf{μ>0:ϱp(·)(1μf)≤1}.$

If p(x) ≡ p is constant, then Lp(·)(ℝn) = Lp(ℝn) is the classical Lebesgue space.

### Definition 1.2

We say that a function g : ℝn → ℝ is locally log-Hölder continuous, if there exists a constant clog > 0 such that

$∣g(x)-g(y)∣≤clogln(e+1/∣x-y∣)$

for all x, y ∈ ℝn. If

$∣g(x)-g(0)∣≤clogln(e+1/∣x∣)$

for all x ∈ ℝn, then we say that g is log-Hölder continuous at the origin (or has a log decay at the origin). If, for some g ∈ ℝ and clog > 0, there holds

$∣g(x)-g∞∣≤clogln(e+∣x∣)$

for all x ∈ ℝn, then we say that g is log-Hölder continuous at infinity (or has a log decay at infinity).

As an example of a function locally log-Hölder continuous, see E. Nakai and Y. Sawano [12, Example 1.3].

The set $P0log(ℝn)$ and $P∞log(ℝn)$ consist of all exponents p ∈ ℘(ℝn) which have a log decay at the origin and at infinity, respectively. The set ℘log(ℝn) is used for all those exponents p ∈ ℘(ℝn) which are locally log-Hölder continuous and have a log decay at infinity, with p := lim|x|→∞p(x).

It is well known that if p ∈ ℘log(ℝn) then p′ ∈ ℘log(ℝn), where p′ denotes the conjugate exponent of p given by 1/p(·) + 1/p′ (·) = 1.

### Definition 1.3

Let p, q ∈ ℘0(ℝn). The mixed Lebesgue-sequence space ℓq(·)(Lp(·)) is defined on sequences of Lp(·)-functions by the modular

$ϱℓq(·) (Lp(·))((fv)v)=∑vinf{λv>0:ϱp(·)(fvλv1/q(·))≤1}.$

The (quasi)-norm is defined from this as usual:

$‖(fv)v‖ℓq(·) (Lp(·))=inf{γ>0:ϱℓq(·) (Lp(·))(1γ(fv)v)≤1}.$

Since q+< ∞, then we can replace by the simpler expression $ϱℓq(·) (Lp(·))((fv)v)=∑v‖∣fv∣q(·)‖p(·)q(·)$.

If E ⊂ ℝn is a measurable set, then |E| stands for the (Lebesgue) measure of E and χE denotes its characteristic function. Before giving the definition of variable Herz spaces, let us introduce the following notations

$Bk:=B(0,2k), Rk:=Bk/Bk-1 and χRk, k∈ℤ.$

### Definition 1.4

Let p, q ∈ ℘0(ℝn) and α : ℝn → ℝ with αL(ℝn). The inhomogeneous Herz space $Kp(·),q(·)α(·)(ℝn)$ consists of all $f∈LLocp(·)(ℝn)$ such that

$‖f‖Kp(·),q(·)α(·):=‖fχB0‖p(·)+‖(2kα(·)fχk)k≥1‖ℓq(·) (Lp(·))<∞.$

Similarly, the homogeneous Herz space $K˙p(·)α(·),q(·)(ℝn)$ is defined as the set of all $f∈LLocp(·)(ℝn/{0})$ such that

$‖f‖K˙p(·)α(·),q(·)(ℝn):=‖(2kα(·)fχk)k∈ℤ‖ℓq(·) (Lp(·))<∞.$

The variable Herz spaces $Kp(·)α(·),q(·)(ℝn)$ and $K˙p(·)α(·),q(·)(ℝn)$, were first introduced by Izuki and Noi in [8]. In [6] the authors obtained a new equivalent norm of these function spaces. We refer the reader to the paper [5, 16] for further results for these function spaces. If α(·), p(·) and q(·) are constants, then $K˙q(·)α(·),p(·)(ℝn)$ is the classical Herz space $K˙qα,p(ℝn)$.

The following proposition is very important for the proof of the main results in this paper; it is from D. Drihem and F. Seghiri in [6].

### Proposition 1.5

Let αL(ℝn), p, q ∈ ℘0(ℝn). If α and q are log-Hölder continuous at infinity, then

$Kp(·)α(·),q(·)(ℝn)=Kp(·)α∞,q∞(ℝn).$

Additionally, if α(·) and q(·) have a log decay at the origin, then

$‖f‖K˙p(·)α(·),q(·)(ℝn)≈(∑k=-∞-1‖2kα(0)fχk‖p(·)q(0))1/q(0)+(∑k=0∞‖2kα∞fχk‖p(·)q∞)1/q∞.$

The Hardy-Littlewood maximal operator ℳ is defined on $Lloc1$ by

$M(f)(x):=supr>01∣B(x,r)∣∫B(x,r)∣f(y)∣dy,$

where B(x, r) is the open ball in ℝn centered at x ∈ ℝn and radius r > 0. It was shown in [4, Theorem 4.3.8] that ℳ : Lp(·)Lp(·) is bounded if p ∈ ℘log and p > 1, see also [3, Theorem 1.2].

Let $ϕ∈C0∞(ℝn)$ with supp ϕB0, ∫nϕ(x)dx ≠ 0 and $ϕt(·)=t-nϕ(·t)$ for any t > 0. Let ℳϕ(f) be the grand maximal function of f defined by

$Mϕ(f) (x):=supt>0∣ϕt*f(x)∣.$

Here we give the definition of the homogeneous Herz-type Hardy spaces $HK˙p(·)α(·),q(·)$.

### Definition 1.6

Let p, q ∈ ℘0(ℝn) and α : ℝn → ℝ with αL(ℝn). The homogeneous Herz-type Hardy space $HK˙p(·)α(·),q(·)(ℝn)$ is defined as the set of all such that $Mϕ(f)∈K˙p(·)α(·),q(·)(ℝn)$ and we define

$‖f‖HK˙p(·)α(·),q(·):=‖Mϕ(f)‖K˙p(·)α(·),q(·).$

It can be shown that if α(·), p(·) and q(·) satisfy the conditions of Definition 1.6, then the quasi-norm $‖f‖HK˙p(·)α(·),q(·)$ does not depend, up to the equivalence of quasi-norms, on the choice of the function ϕ and, hence, the space $HK˙p(·)α(·),q(·)(ℝn)$ is defined independently of the choice of ϕ. If $p∈P0log(ℝn)∩P∞log(ℝn)$ with $-np+<α-≤α+ and q ∈ ℘0(ℝn) then $HK˙p(·)α(·),q(·)(ℝn)=K˙p(·)α(·),q(·)(ℝn)$. If α (·) = 0, p (·) = q (·) then $HK˙p(·)α(·),q(·)(ℝn)$ and $K˙p(·)α(·),q(·)(ℝn)$ coincide with Lp(·)(ℝn).

One recognizes immediately that if α(·), p(·) and q(·) are constants, then the spaces $HK˙pα,q$ are just the usual Herz-type Hardy spaces were recently studied in [10] and [11].

We refer the reader to the recent monograph [4, section 4.5] for further details, historical remarks and more references on variable exponent spaces.

2. Some Technical Lemmas

In this section, we present six lemmas used to prove our main theorems in Section 3. Recall that the expression fg means that fcg for some independent constant c (and non-negative functions f and g), and fg means fgf.

Lemma 2.1 plays an important role in the proof of main results; Lemma 2.2 is a Hardy-type inequality which is easy to prove; Lemma 2.3 presents the Hölder inequality in Lp(·)(ℝn); Lemma 2.4 presents the Lp(·)-boundedness of μ; Lemma 2.5 treats the boundedness of fractional integral on variable Lebesgue space; and the last Lemma presents the boundedness of homogeneous function of degree zero.

### Lemma 2.1.([1])

Let $p∈P∞log(ℝn)$and let $R=B(0,r)/B(0,r2)$. If |R| ≥ 2n, then

$‖χR‖p(·)≈∣R∣1p(x)≈∣R∣1p∞$

with the implicit constants independent of r and xR.

The left-hand side equivalence remains true for every |R| > 0 if we assume, additionally, $p∈P0log(ℝn)∩P∞log(ℝn)$.

### Lemma 2.2.([5])

Let 0 < a < 1 and 0 < q ≤ ∞. Let {ɛk}k∈ℤbe a sequence of positive real numbers, such that

$‖{ɛk}k∈ℤ‖ℓq=I<∞.$

Then the sequences {δk : δk = ∑jk akjɛj}k∈ℤand {ηk : ηk = ∑jk ajkɛj}k∈ℤbelong to ℓq, and

$‖{δk}k∈ℤ‖ℓq+‖{ηk}k∈ℤ‖ℓq≤cI,$

with c > 0 only depending on a and q.

### Lemma 2.3.([4])

Let p ∈ ℘(ℝn). Then for all fLp(·)(ℝn) and gLp′(·)(ℝn), fgL1(ℝn) and

$‖fg‖1≤2‖f‖p(·)‖g‖p′(·).$

### Lemma 2.4.([9])

Let p ∈ ℘log(ℝn), then there exists a constant C such that for any fLp(·)(ℝn)

$‖μ(f)‖p(·)≤C‖f‖p(·).$

### Lemma 2.5.([17])

Suppose that p1, p2 ∈ ℘log(ℝn) with $p1+and $1p1(·)-1p2(·)=σn$. Then for all fLp1(·) (ℝn), we have

$‖∫ℝnf(y)∣·-y∣n-σdy‖p2(·)≤c‖f‖p1(·).$

### Lemma 2.6.([14])

If a > 0, 1 ≤ s ≤ ∞, 0 < ds andn + (n − 1)d/s < τ < ∞, then

$(∫∣y∣≤a∣x∣∣y∣τ∣Ω(x-y)∣ddy)1d≤c∣x∣(τ+n/d)‖Ω‖Ls(Sn-1).$
3. Variable Herz Estimate of Marcinkiewicz Integral Operators

In this section, we present two results concerning the Marcinkiewicz integral operator μ. In the first, we show that μ is bounded from $K˙p(·)α(·),q(·)(ℝn)$ to $K˙p(·)α(·),q(·)(ℝn)$ for α(·), p(·) and q(·) satisfies some conditions. Next, we present the boundedness of μ on variable Herz-type Hardy spaces $HK˙p(·)α(·),q(·)(ℝn)$.

Motivated by [9] and [14], we generalize the boundedness for Marcinkiewicz integral operators μ to the case of variable Herz spaces ( all exponents are variables). One of our main results can be stated as follows.

### Theorem 3.1

Suppose that 0 < τ ≤ 1, p ∈ ℘log(ℝn) with p+< ∞, Ω ∈ Ls(Sn−1), s > (p′)and αL(ℝn), q ∈ ℘0(ℝn). If α and q have a log decay at the origin such that

$-np(0)-ns-τ<α(0)

then μ is bounded from $K˙p(·)α(·),q(·)(ℝn)$(or $Kp(·)α(·),q(·)(ℝn)$) to $K˙p(·)α(·),q(·)(ℝn)$(or $Kp(·)α(·),q(·)(ℝn)$).

### Remark 3.2

We would like to mention if α(·) and q(·) are constants, then the statements corresponding to Theorem 3.1 can be found in Theorem 2.1 of [14].

In the following, we use c as a generic positive constant, i.e. a constant whose value may change from appearance to appearance.

Proof of Theorem 3.1

We show that

$‖μ(f)‖K˙p(·)α(·),q(·)(ℝn)≤c‖f‖K˙p(·)α(·),q(·)(ℝn)$

for all $f∈K˙p(·)α(·),q(·)(ℝn)$. Using Proposition 1.5, we have μ(f) in $K˙p(·)α(·),q(·)(ℝn)$-norm is equivalent to

${∑k=-∞-12kα(0)q(0)‖μ(f)χk‖p(·)q(0)}1/q(0)+{∑k=0+∞2kα∞(q)∞‖μ(f)χk‖p(·)(q)∞}1/(q)∞,$

we write

$f=∑j∈ℤfχj=∑j∈ℤfj,$

then

$‖μ(f)‖K˙p(·)α(·),q(·)(ℝn)≲{∑k=-∞-12kα(0)q(0)(∑j=-∞k-2‖μ(fj)χk‖p(·))q(0)}1/q(0)+{∑k=-∞-12kα(0)q(0)(∑j=k-2k+1‖μ(fj)χk‖p(·))q(0)}1/q(0)+{∑k=-∞-12kα(0)q(0)(∑j=k+2∞‖μ(fj)χk‖p(·))q(0)}1/q(0)+{∑k=0∞2kα∞q∞(∑j=-∞k-2‖μ(fj)χk‖p(·))q∞}1/q∞+{∑k=0∞2kα∞q∞(∑j=k-2∞‖μ(fj)χk‖p(·))q∞}1/q∞+{∑k=0∞2kα∞q∞(∑j=k+2∞‖μ(fj)χk‖p(·))q∞}1/q∞=:H1+H2+H3+H4+H5+H6.$

Let us estimate H1 and H4. We consider

$∣μfj(x)∣≤(∫0∣x∣|∫∣x-y∣≤tΩ(x-y)∣x-y∣n-1fj(y) dy|2dtt3)12+(∫∣x∣∞|∫∣x-y∣≤tΩ(x-y)∣x-y∣n-1fj(y) dy|2dtt3)12:=I1+I2.$

Observe that in this case xRk, yRj and jk − 2. So we know that |xy| ≈ |x| ≈ 2k, and by mean value theorem, we have

$|1∣x-y∣2-1x2|≤c∣y∣∣x-y∣3.$

By (3.1), the Minkowski inequality and the generalized Hölder inequality, we have

$I1≲∫ℝn∣Ω(x-y)∣∣x-y∣n-1∣fj(y)∣|∫∣x-y∣∣x∣dtt3|12dy≲∫ℝn∣Ω(x-y)∣∣x-y∣n-1∣fj(y)∣∣y∣12∣x-y∣32dy≲2j/2∣x∣n+1/2∫ℝn∣Ω(x-y)∣∣fj(y)∣dy≲2-nk‖fj‖p(·)‖Ω(x-·)χj‖p′(·).$

The estimation of I2 is the same as before since we never use |xy| ≈ |x|, then we obtain

$I2≲2-nk‖fj‖p(·)‖Ω(x-·)χj‖p′(·).$

We can obtain that each term (I1 and I2) is no more than

$c2-nk‖fj‖p(·)‖Ω(x-·)χj‖p′(·),$

in the other hand, by Hölder inequality and Lemma 2.6, we obtain

$‖Ω(x-·)χj‖p′(·)≤‖Ω(x-·)χj‖s‖χj‖θ(·)≤2-jτ(∫Rj∣y∣sτ∣Ω(x-y)∣sdy)1s‖χj‖θ(·)≲2-jτ2k(τ+ns)‖Ω‖Ls(Sn-1)‖χj‖θ(·)$

where $1p′(·)=1s+1θ(·)$. Since $θ∈P∞log(ℝn)$, we have for any j

$‖χj‖θ(·)≈‖χj‖p′(·)∣Rj∣-1s,$

which gives

$‖μ(fj)χk‖p(·)≲2-nk2-(ns+τ) (j-k)‖fj‖p(·)‖χj‖p′(·)‖χk‖p(·)‖Ω‖Ls(Sn-1).$

Estimation of H1. In this case, since k and j are negative integers, by Lemma 2.1 and since Ω ∈ Ls(Sn−1), we have

$‖μ(fj)χk‖p(·)≲2(n-np(0)-τ-ns) (j-k)‖fj‖p(·) ‖Ω‖Ls(Sn-1)≲2(n-np(0)-τ-ns) (j-k)‖fj‖p(·),$

which gives

$H1≲{∑k=-∞-12kα(0)q(0)(∑j=-∞k-22(j-k) (n-np(0)-τ-ns)q(0)‖fj‖p(·)q(0))}1/q(0)=c{∑k=-∞-1(∑j=-∞k-22(j-k) (n-α(0)-np(0)-τ-ns)q(0)(2jα(0)q(0)‖fj‖p(·)q(0)))}1/q(0)$

since $α(0)-n+np(0)-ns-τ<0$, then by Lemma 2.2, we have

$H1≤c{∑j=-∞-12jα(0)q(0)‖fj‖p(·)q(0)}1/q(0)≤c‖f‖K˙p(·)α(·),q(·)(ℝn).$

Estimation of H4. We split

$∑j=-∞k-2‖μ(fj)χk‖p(·)=∑j=-∞-1⋯+∑j=0k-2⋯$

then H4 can be estimated by

$H41+H42,$

where

$H41:={∑k=0∞2kα∞q∞(∑j=-∞-1‖μ(fj)χk‖p(·))q∞}1/q∞$

and

$H42:={∑k=0∞2kα∞q∞(∑j=0k-2‖μ(fj)χk‖p(·))q∞}1/q∞$

Let estimate $H41$. We have in this case j < 0 ≤ k. By Lemma 2.1, we obtain

$‖μ(fj)χk‖p(·)≤c2k(np∞-n+ns+τ)2j(n-ns-τ-np(0))‖fj‖p(·)$

therefore, $H41$ is bounded by

$csupj≤02α(0)‖fj‖p(·){∑k=0∞2k(α∞+np∞-n+ns+τ)q∞(∑j=-∞-12j(n-ns-τ-np(0)))q∞}1/q∞,$

by embedding q(0) and since $α∞+np∞-n+ns+τ<0, we have

$H41≤c(∑j=-∞-12α(0)q(0)j‖fj‖p(·)q(0))1/q(0)≤c‖f‖K˙p(·)α(·),q(·)(ℝn).$

We can estimate $H42$ by the same argument used in the estimation of H1 if we replace α(0), p(0) and q(0) by α, p and q respectively, then $H42$ is bounded by

${∑k=0∞(∑j=0k-22(j-k) (n-np∞-τ-ns-α∞)(2jα∞‖fj‖p(·)))q∞}1/q∞≤c(∑k=0∞2jα∞q∞‖fj‖p(·)q∞)1/q∞≤c‖f‖K˙p(·)α(·),q(·)(ℝn)$

Let us estimate H2+H5. By the (Lp(·) (ℝn), Lp(·) (ℝn))-boundedness of μ, we have

$H2+H5≲(∑k=-∞-1‖2kα(0)fχk‖p(·)q(0))1/q(0)+(∑k=0∞‖2kα∞fχk‖p(·)q∞)1/q∞≲‖f‖K˙p(·)α(·),q(·)(ℝn).$

Let us estimate H3. It is possible to prove the following estimation (similar to the estimate for I1 and I2)

$∣μ(fj)∣≤c2-nj‖fj‖p(·)‖Ω(x-·)χj‖p′(·),$

for the detailed proof of this estimation, see [14, p.259–260]. By Hölder inequality and Lemma 2.6, the right-hand of (3.4) is bounded by

$c2-jτ(∫Rj∣y∣sτ∣Ω(x-y)∣sdy)1s‖χj‖θ(·)≲2-nj2k(τ+ns)‖χj‖p′(·)‖Ω‖Ls(Sn-1),$

which gives

$‖μ(fj)χk‖p(·)≲2-nj2(k-j) (τ+ns)‖fj‖p(·)‖χj‖p′(·)‖χk‖p(·).$

We split

$∑j=k+2∞‖μ(fj)χk‖p(·)=∑j=k+2-1⋯+∑j=0∞⋯$

Then H3 is bounded by

${∑k=-∞-12kα(0)q(0)(∑j=k+2-1‖μ(fj)χk‖p(·))q(0)}1/q(0)+{∑k=-∞-12kα(0)q(0)(∑j=0∞‖μ(fj)χk‖p(·))q(0)}1/q(0)=:H31+H32$

For $H31$, since j and k are negative integers, we have

$H31≲{∑k=-∞-1(∑j=k+2-12(α(0)+np(0)+τ+ns) (k-j)2jα(0)‖fj‖p(·))q(0)}1/q(0)$

Since $α(0)+np(0)+τ+ns>0$, by Lemma 2.2, we obtain

$H31≲(∑k=-∞-12kα(0)q(0)‖fχk‖p(·)q(0))1/q(0)≲‖f‖K˙p(·)α(·),q(·)(ℝn).$

For $H32$, since k < 0 ≤ j, then we have

$‖μ(fj)χk‖p(·)≤c2(k-j) (τ+ns)‖fj‖p(·)2-njp∞2nkp(0)≤c2k(τ+ns+np(0))2jα∞‖fj‖p(·)2-j(np∞+α∞+τ+ns),$

by Hölder inequality in 1 and since $γ=np∞+α∞+τ+ns>0$, we have

$H32≲{∑k=-∞-12k(τ+ns+np(0))(∑j=0∞2jα∞‖fj‖p(·)2-jγ)q(0)}1/q(0)≲(∑k=-∞-12kηq(0))1/q(0)(∑j=0∞2-jγq∞′)1/q∞′(∑j=0∞2jα∞q∞‖fj‖p(·)q∞)1/q∞≲(∑j=0∞2jα∞q∞‖fj‖p(·)q∞)1/q∞≲‖f‖K˙p(·)α(·),q(·)(ℝn),$

where $η=τ+ns+np(0)>0$.

Let us estimate H6. In this case, since k and j are non-negative integers, by (3.3) and Lemma 2.1, we have

$‖μ(fj)χk‖p(·)≤c2(-n+np∞+τ+ns) (k-j)‖fj‖p(·),$

which gives

$H6≲{∑k=0∞(∑j=k+2∞2(k-j) (n-γ)q∞(2jα∞q∞‖fj‖p(·)q∞))}1/q∞,$

since nγ > 0, by Lemma 2.2, we obtain

$H6≤{∑j=0∞2jα∞q∞‖fj‖p(·)q∞}1/q∞≤c‖f‖K˙p(·)α(·),q(·)(ℝn).$

### Remark 3.3

A non-homogeneous counterpart of Theorem 3.1 is available. Since $Kp(·)α(·),q(·)(ℝn)=Kp(·)α∞,q∞(ℝn)$, their proof is an immediate consequence of [14, Theorem 2.1].

To prove the Theorem 3.6, we need the notation of atomic decomposition.

### Definition 3.4

Let αL(ℝn), p ∈ ℘(ℝn), q ∈ ℘0(ℝn) and m ∈ ℕ0. A function a is said to be a central (α(·), p(·))-atom, if

• $suppa⊂B(0,r)¯={x∈ℝn:∣x∣≤r}$, r > 0.

• $‖a‖p(·)≤∣B(0,r)¯∣-α(0)/n$, 0 < r < 1.

• $‖a‖p(·)≤∣B(0,r)¯∣-α∞/n$, r ≥ 1.

• nxβa(x)dx = 0, |β| ≤ m.

A function a on ℝn is said to be a central (α(·), p(·))-atom of restricted type, if it satisfies the conditions (iii), (vi) above and suppaB(0, r), r ≥ 1.

If r = 2k for some k ∈ ℤ in Definition 3.4, then the corresponding central (α(·), p(·))-atom is called a dyadic central (α(·), p(·))-atom.

The following theorem presents the atomic decomposition characterization of variable Herz-type Hardy spaces, see [6].

### Theorem 3.5

Let α and q are be log-Hölder continuous, both at the origin and at infinity and p ∈ ℘log(ℝn) with 1 < pp+<. For any $f∈HK˙p(·)α(·),q(·)(ℝn)$, we have

$f=∑k=-∞∞λkak,$

where the series converges in the sense of distributions, λk ≥ 0, each ak is a central (α(·), p(·))-atom with suppaBk and

$(∑k=-∞-1∣λk∣q(0))1/q(0)+(∑k=0∞∣λk∣q∞)1/q∞≤c‖f‖HK˙p(·)α(·),q(·).$

Conversely, if $α(·)≥n(1-1p-)$and $m≥[α++n(1p--1)]$, and if holds, then $f∈HK˙p(·)α(·),q(·)(ℝn)$, and

$‖f‖HK˙p(·)α(·),q(·)≈inf{(∑k=-∞-1∣λk∣q(0))1/q(0)+(∑k=0∞∣λk∣q∞)1/q∞},$

where the infimum is taken over all the decompositions of f as above.

In the next result we treat the boundedness of Marcinkiewicz integral operators with homogeneous kernel on variable Herz-type Hardy spaces.

### Theorem 3.6

Suppose that p1, p2 ∈ ℘log(ℝn) with $p1+<2n$and $1p1(·)-1p2(·)=12n$, αL(ℝn), q1, q2 ∈ ℘0(ℝn),Ω ∈ Ls(Sn−1) with $s>(p1′)-$. If α, q1and q2are log-Hölder continuous, both at the origin and at infinity such that

$α(·)≥n(1-1p1-),q1(0)≤q2(0) and (q1)∞≤(q2)∞.$

Then μ is bounded from $HK˙p1(·)α(·),q1(·)(ℝn)$to $K˙p2(·)α(·),q2(·)(ℝn)$.

Proof

We must show that

$‖μ(f)‖K˙p2(·)α(·),q2(·)(ℝn)≤c‖f‖HK˙p1(·)α(·),q1(·)(ℝn)$

for all $f∈HK˙p1(·)α(·),q1(·)(ℝn)$. Using Theorem 3.5, we may assume that

$f=∑i=-∞+∞λiai$

where λi ≥ 0 and ai’s are (α (·), p1 (·))-atom with suppaiBi. Using Proposition 1.5, we have

$‖μ(f)‖K˙p2(·)α(·),q2(·)(ℝn)≈{∑k=-∞-12kα(0)q2(0)‖μ(f)χk‖p2(·)q2(0)}1/q2(0)+{∑k=-∞-12kα∞(q2)∞‖μ(f)χk‖p2(·)(q2)∞}1/(q2)∞≤{∑k=-∞-12kα(0)q2(0)(∑i=-∞k-3∣λi∣‖μ(ai)χk‖p2(·))q2(0)}1/q2(0)+{∑k=-∞-12kα(0)q2(0)(∑i=k-2∞∣λi∣‖μ(ai)χk‖p2(·))q2(0)}1/q2(0)+{∑k=0+∞2kα∞(q2)∞(∑i=-∞k-3∣λi∣‖μ(ai)χk‖p2(·))(q2)∞}1/(q2)∞+{∑k=0+∞2kα∞(q2)∞(∑i=k-2+∞∣λi∣‖μ(ai)χk‖p2(·))(q2)∞}1/(q2)∞=:F1+F2+F3+F4.$

Let us estimate F1. We consider

$∣μ(ai) (x)∣≤(∫0∣x∣|∫∣x-y∣≤tΩ(x-y)∣x-y∣n-1ai(y) dy|2dtt3)12+(∫∣x∣∞|∫∣x-y∣≤tΩ(x-y)∣x-y∣n-1ai(y) dy|2dtt3)12:=Q1+Q2.$

Observe that in this case xRk, yBi and ik − 3. So we know that |xy| ≈ |x| ≈ 2k. By (3.2), we have

$Q1≲∫Bi∣Ω(x-y)∣∣x-y∣n-1∣y∣12∣x-y∣32∣ai(y)∣dy,$

by the m-order vanishing moments of ai with $m≥[α+-n(1-1p1-)]$, we can subtract the Taylor expansion of $∣x-y∣12-n$ at x, we obtain

$Q1≤∫Bi∣y∣m+1∣x∣n+m+12∣ai(y)∣∣Ω(x-y)∣dy≤c2-k(n+m+12)+i(m+1)∫Bi∣ai(y)∣∣Ω(x-y)∣dy,$

The estimation of Q2 is the same as before since we never use |xy| ≈ |x|, we have

$∣μ(ai) (x)∣≤c2-k(n+m+12)+i(m+1)∫Bi∣ai(y)∣∣Ω(x-y)∣ dy,$

as the same reason in the proof of (3.3), we get

$‖μ(ai)χk‖p2(·)≲2-(n-12)k2β(i-k)‖ai‖p1(·)‖χi‖p1′(·)‖χk‖p2(·)‖Ω‖Ls(Sn-1)≲2-(n-12)k2β(i-k)‖ai‖p1(·)‖χi‖p1′(·)‖χk‖p2(·),$

where $β=(1+m-ns-τ)$.

On the other hand (see [7, p.350] for $σ=12$), we have

$∫BkχBk(y)∣x-y∣n-12dy≥∫Bkdy∣x-y∣n-12χBk(x)≥c2k2χBk(x).$

(3.5), (3.6) and Lemma 2.5, gives

$‖μ(ai)χk‖p2(·)≤2-(n-12)k2β(i-k)‖ai‖p1(·)‖χBi‖p1′(·)‖χk‖p2(·)≤2-nk2β(i-k)‖ai‖p1(·)‖χBi‖p1′(·)‖∫BkχBk(y)dy∣·-y∣n-12‖p2(·)≤2-nk2β(i-k)‖ai‖p1(·)‖χBi‖p1′(·)‖χBk‖p1(·),$

by Lemma 2.1, we have

$F1={∑k=-∞-12kα(0)q2(0)(∑i=-∞k-3∣λi∣‖μ(ai)χk‖p2(·))q2(0)}1/q2(0)≤c{∑k=-∞-1(∑i=-∞k-3∣λi∣2(i-k) (β-(α+n/p1) (0)))q2(0)}1/q2(0),$

since we can choose m large enough such that $β-α++n(1-1p1-)>0$, by Lemma 2.2, we obtain

$F1≤c(∑k=-∞-1∣λk∣q2(0))1/q2(0)≤c(∑k=-∞-1∣λk∣q1(0))1/q1(0)≤c‖f‖HK˙p1(·)α(·),q1(·).$

Let us estimate F2. By Lemma 2.4 and applying the size condition of ai (conditions (ii) and (iii) in Definition 3.4), we have

$F2={∑k=-∞-12kα(0)q2(0)(∑i=k-2+∞∣λi∣‖μ(ai)χk‖p2(·))q2(0)}1/q2(0)≤c{∑k=-∞-12kα(0)q2(0)(∑i=k-2+∞∣λi∣‖ai‖p2(·))q2(0)}1/q2(0)≤c{∑k=-∞-12kα(0)q2(0)(∑i=k-2-1∣λi∣‖ai‖p2(·))q2(0)}1/q2(0)+c{∑k=-∞-12kα(0)q2(0)(∑i=0+∞∣λi∣‖ai‖p2(·))q2(0)}1/q2(0)≤c{∑k=-∞-1(∑i=k-2-1∣λi∣2(k-i)α(0))q2(0)}1/q2(0)+c{∑k=-∞-1(∑i=0+∞∣λi∣2(k-i)α-+k(α(0)-α-)+i(α--α∞))q2(0)}1/q2(0)$

for k < 0 ≤ i and since α ≤ min(α (0), α), we have

$k(α(0)-α-)+i(α--α∞)≤0.$

By Lemma 2.2, we obtain

$F2≤c(∑k=-∞-1∣λk∣q2(0))1/q2(0)≤c(∑k=-∞-1∣λk∣q1(0))1/q1(0)≤c‖f‖HK˙p1(·)α(·),q1(·).$

We can estimate F3 and F4 by the same arguments used in the estimation of F1 and F2 if we replace α(0), p2(0) and q2(0) by α, (p2) and (q2) respectively.

A combination of estimations of F1, F2, F3 and F4 completes the proof of Theorem 3.6.

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