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KYUNGPOOK Math. J. 2019; 59(2): 259-275

Published online June 23, 2019

Copyright © Kyungpook Mathematical Journal.

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

Received: March 22, 2018; Revised: November 2, 2018; Accepted: December 6, 2018

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.

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(n):={f:fLipβ(n)=supx,yn;xyf(x)-f(y)x-yβ<}.

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):=(0FΩf(x)2dtt3)12

where

FΩf(x):=x-ytΩ(x-y)x-yn-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(·)(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

  1. A. Almeida, and D. Drihem. Maximal, potential and singular type operators on Herz spaces with variable exponents. J Math Anal Appl., 394(2012), 781-795.
    CrossRef
  2. A. Benedek, A-P. Calderón, and R. Panzone. Convolution operators on Banach space valued functions. Proc Nat Acad Sci USA., 48(1962), 356-365.
    Pubmed KoreaMed CrossRef
  3. L. Diening, P. Harjulehto, P. Hasto, Y. Mizuta, and T. Shimomura. Maximal functions in variable exponent spaces: limiting cases of the exponent. Ann Acad Sci Fenn Math., 34(2009), 503-522.
  4. L. Diening, P. Harjulehto, P. Hasto, and M. Ruzicka. Lebesgue and sobolev spaces with variable exponents. Lecture Notes in Mathematics, 2017, Springer Verlag, Berlin, 2011.
    CrossRef
  5. D. Drihem, and R. Heraiz. Herz type Besov spaces of variable smoothness and integrability. Kodai Math J., 40(2017), 31-57.
    CrossRef
  6. D. Drihem, and F. Seghiri. Notes on the Herz-type Hardy spaces of variable smoothness and integrability. Math Inequal Appl., 19(2016), 145-165.
    CrossRef
  7. M. Izuki. Fractional integrals on Herz–Morrey spaces with variable exponent. Hiroshima Math J., 40(2010), 343-355.
    CrossRef
  8. M. Izuki, and T. Noi. Boundedness of some integral operators and commutators on generalized Herz spaces with variable exponents, , OCAMI Preprint Series, 2011–15.
  9. Z. Liu, and H. Wang. Boundedness of Marcinkiewicz integrals on Herz spaces with variable exponent. Jordan J Math Stat., 5(2012), 223-239.
  10. S. Lu, and D. Yang. Some characterizations of weighted Herz-type Hardy spaces and their applications. Acta Math Sinica (NS)., 13(1997), 45-58.
    CrossRef
  11. A. Miyachi. Remarks on Herz-type Hardy spaces. Acta Math Sinica (Engl Ser)., 17(2001), 339-360.
    CrossRef
  12. E. Nakai, and Y. Sawano. Hardy spaces with variable exponents and generalized Campanato spaces. J Funct Anal., 262(2012), 3665-3748.
    CrossRef
  13. EM. Stein. On the functions of Littlewood–Paley, Lusin, and Marcinkiewicz. Trans Amer Math Soc., 88(1958), 430-466.
    CrossRef
  14. H. Wang. Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent. Czech Math J., 66(2016), 251-269.
    CrossRef
  15. H. Wang. The continuity of commutators on Herz-type Hardy spaces with variable exponent. Kyoto J Math., 56(2016), 559-573.
    CrossRef
  16. HB. Wang, and ZG. Liu. The Herz-type Hardy spaces with variable exponent and their applications. Taiwanese J Math., 16(2012), 1363-1389.
    CrossRef
  17. H. Wang, L. Zongguang, and F. Zunwei. Boundedness of Fractional Integrals on Herztype Hardy Spaces with variable exponent. Adv Math (China)., 46(2017), 252-260.
  18. L. Xu. Boundedness of Marcinkiewicz integral on weighted Herz-type Hardy spaces. Anal Theory Appl., 22(2006), 56-64.
    CrossRef