Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces

Kristóf Szarvas, F. Weisz

Research output: Contribution to journalArticle

Abstract

The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces Lp(Rd) (in the case p > 1), but (in the case when 1/p(·) is log-Hölder continuous and p- = inf{p(x): x ∈ Rd > 1) on the variable Lebesgue spaces Lp(·)(Rd), too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type (1, 1). In the present note we generalize Besicovitch’s covering theorem for the so-called γ-rectangles. We introduce a general maximal operator Ms γδ, and with the help of generalized Φ-functions, the strong- and weak-type inequalities will be proved for this maximal operator. Namely, if the exponent function 1/p(·) is log-Hölder continuous and p- ≥ s, where 1 ≤ s ≤ ∞ is arbitrary (or p- ≥ s), then the maximal operator Ms γδ is bounded on the space Lp(·)(Rd) (or the maximal operator is of weak-type (p(·), p(·))).

Original languageEnglish
Pages (from-to)1079-1101
Number of pages23
JournalCzechoslovak Mathematical Journal
Volume66
Issue number4
DOIs
Publication statusPublished - Dec 1 2016

    Fingerprint

Keywords

  • Besicovitch’s covering theorem
  • maximal operator
  • strong-type inequality
  • variable Lebesgue space
  • weak-type inequality
  • γ-rectangle

ASJC Scopus subject areas

  • Mathematics(all)

Cite this