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The Hardy--Littlewood maximal operator and BLO1/log class of exponents

  • Tengiz Kopaliani and Shalva Zviadadze EMAIL logo
Published/Copyright: July 21, 2016
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Georgian Mathematical Journal
From the journal Georgian Mathematical Journal Volume 23 Issue 3

Abstract

It is well known that if the Hardy–Littlewood maximal operator is bounded in the variable exponent Lebesgue space Lp⁢(⋅)⁢[0;1], then p⁢(⋅)∈BMO1/log. On the other hand, there exists an exponent p⁢(⋅)∈BMO1/log, 1<p-≤p+<∞, such that the Hardy–Littlewood maximal operator is not bounded in Lp⁢(⋅)⁢[0;1]. But for any exponent p⁢(⋅)∈BMO1/log, 1<p-≤p+<∞, there exists a constant c>0 such that the Hardy–Littlewood maximal operator is bounded in Lp⁢(⋅)+c⁢[0;1]. In this paper, we construct an exponent p⁢(⋅), 1<p-≤p+<∞, 1/p⁢(⋅)∈BLO1/log such that the Hardy–Littlewood maximal operator is not bounded in Lp⁢(⋅)⁢[0;1].

Keywords: Variable exponent Lebesgue space; Hardy–Littlewood maximal operator
MSC 2010: 42B25; 42B35

Funding source: Shota Rustaveli National Science Foundation

Award Identifier / Grant number: 31/48

Award Identifier / Grant number: DI/9/5-100/13

Award Identifier / Grant number: 52/36

Funding statement: The research of the first author is supported by Shota Rustaveli National Science Foundation grants no. 31/48 (Operators in some function spaces and their applications in Fourier Analysis) and no. DI/9/5-100/13 (Function spaces, weighted inequalities for integral operators and problems of summability of Fourier series). The research of the second author is supported by Shota Rustaveli National Science Foundation grant no. 52/36.

References

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Received: 2014-12-23
Accepted: 2015-12-22
Published Online: 2016-7-21
Published in Print: 2016-9-1

© 2016 by De Gruyter

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  2. Non-trivial solutions for nonlocal elliptic problems of Kirchhoff-type
  3. Existence of renormalized solutions for strongly nonlinear parabolic problems with measure data
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  5. A spectral representation of the linear multivelocity transport problem
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https://doi.org/10.1515/gmj-2016-0032
Keywords for this article
Variable exponent Lebesgue space; Hardy–Littlewood maximal operator

Articles in the same Issue

  1. Frontmatter
  2. Non-trivial solutions for nonlocal elliptic problems of Kirchhoff-type
  3. Existence of renormalized solutions for strongly nonlinear parabolic problems with measure data
  4. On the modification of the Szaśz–Durrmeyer operators
  5. A spectral representation of the linear multivelocity transport problem
  6. A Tauberian theorem for the product of Abel and Cesàro summability methods
  7. The spaces of bilinear multipliers of weighted Lorentz type modulation spaces
  8. Summations of Schlömilch series containing Struve function terms
  9. On nondifferentiable minimax semi-infinite programming problems in complex spaces
  10. Modified relativistic Laguerre polynomials. Monomiality and Lie algebraic methods
  11. On the difference between a Vitali–Bernstein selector and a partial Vitali–Bernstein selector
  12. The Hardy--Littlewood maximal operator and BLO1/log class of exponents
  13. Bicritical domination and double coalescence of graphs
  14. The asymptotic behavior of a counting process in the max-scheme. A discrete case
  15. Functions of bounded fractional differential variation – A new concept
  16. Sensitivity analysis of the optimal exercise boundary of the American put option
  17. Estimates for the asymptotic convergence of a non-isothermal linear elasticity with friction
  18. A coupled system of nonlinear differential equations involving m nonlinear terms
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