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 : and
* Corresponding Author.
Received: March 22, 2018; Revised: November 2, 2018; Accepted: December 6, 2018; Published online: June 23, 2019.
© Kyungpook Mathematical Journal. All rights reserved.

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The aim of this paper is to prove that Marcinkiewicz integral operators are bounded from K˙p(·)α(·),q(·)(n) to K˙p(·)α(·),q(·)(n) 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(·)(n).

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


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




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(·)(n) and variable Herz-type Hardy spaces HK˙p(·)α(·),q(·)(n). We define the set of variable exponents by

P0(n):={pmeasurable:p(·):n[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-=essinfxnp(x),         p+esssupxnp(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


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


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


for all x, y ∈ ℝn. If


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


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(n) and Plog(n) 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


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


Since q+< ∞, then we can replace by the simpler expression ϱq(·)(Lp(·))((fv)v)=vfvq(·)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.

Definition 1.4

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


Similarly, the homogeneous Herz space K˙p(·)α(·),q(·)(n) is defined as the set of all fLLocp(·)(n/{0}) such that


The variable Herz spaces Kp(·)α(·),q(·)(n) and K˙p(·)α(·),q(·)(n), 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(·)(n) is the classical Herz space K˙qα,p(n).

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


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


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


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 ϕC0(n) 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


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(·)(n) is defined as the set of all such that Mϕ(f)K˙p(·)α(·),q(·)(n) and we define


It can be shown that if α(·), p(·) and q(·) satisfy the conditions of Definition 1.6, then the quasi-norm fHK˙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(·)(n) is defined independently of the choice of ϕ. If pP0log(n)Plog(n) with -np+<α-α+<n-np- and q ∈ ℘0(ℝn) then HK˙p(·)α(·),q(·)(n)=K˙p(·)α(·),q(·)(n). If α (·) = 0, p (·) = q (·) then HK˙p(·)α(·),q(·)(n) and K˙p(·)α(·),q(·)(n) 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 pPlog(n)and let R=B(0,r)/B(0,r2). If |R| ≥ 2n, then


with the implicit constants independent of r and xR.

The left-hand side equivalence remains true for every |R| > 0 if we assume, additionally, pP0log(n)Plog(n).

Lemma 2.2.([5])

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


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


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


Lemma 2.4.([9])

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


Lemma 2.5.([17])

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


Lemma 2.6.([14])

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

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(·)(n) to K˙p(·)α(·),q(·)(n) 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(·)(n).

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)<n-np(0)-ns-τ   and   -np-ns-τ<α<n-np-ns-τ

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

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


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


we write




Let us estimate H1 and H4. We consider


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


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


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


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


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


where 1p(·)=1s+1θ(·). Since θPlog(n), we have for any j


which gives


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)χkp(·)2(n-np(0)-τ-ns)(j-k)fjp(·)   ΩLs(Sn-1)2(n-np(0)-τ-ns)(j-k)fjp(·),

which gives


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


Estimation of H4. We split


then H4 can be estimated by






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


therefore, H41 is bounded by


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


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


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


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


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


which gives


We split


Then H3 is bounded by


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


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


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


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


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


which gives


since nγ > 0, by Lemma 2.2, we obtain


Remark 3.3

A non-homogeneous counterpart of Theorem 3.1 is available. Since Kp(·)α(·),q(·)(n)=Kp(·)α,q(n), 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

  • suppaB(0,r)¯={xn:xr}, r > 0.

  • ap(·)B(0,r)¯-α(0)/n, 0 < r < 1.

  • ap(·)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 fHK˙p(·)α(·),q(·)(n), we have


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


Conversely, if α(·)n(1-1p-)and m[α++n(1p--1)], and if holds, then fHK˙p(·)α(·),q(·)(n), and


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+<2nand 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(·)(n)to K˙p2(·)α(·),q2(·)(n).


We must show that


for all fHK˙p1(·)α(·),q1(·)(n). Using Theorem 3.5, we may assume that


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


Let us estimate F1. We consider


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


by the m-order vanishing moments of ai with m[α+-n(1-1p1-)], we can subtract the Taylor expansion of x-y12-n at x, we obtain


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


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


where β=(1+m-ns-τ).

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


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


by Lemma 2.1, we have


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


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


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


By Lemma 2.2, we obtain


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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