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# Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

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Volume 68, Issue 5

# On the regularity of one-sided fractional maximal functions

Feng Liu
• College of Mathematics and Systems Science Shandong University of Science and Technology Qingdao Shandong 266590 CHINA
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Published Online: 2018-10-20 | DOI: https://doi.org/10.1515/ms-2017-0171

## Abstract

In this paper we investigate the regularity properties of one-sided fractional maximal functions, both in continuous case and in discrete case. We prove that the one-sided fractional maximal operators ${ℳ}_{\beta }^{+}$ and ${ℳ}_{\beta }^{-}$ map ${W}^{1,p}\left(ℝ\right)$ into ${W}^{1,q}\left(ℝ\right)$ with 1 <p <∞, 0≤β<1/p and q=p/(1-), boundedly and continuously. In addition, we also obtain the sharp bounds and continuity for the discrete one-sided fractional maximal operators ${M}_{\beta }^{+}$ and ${M}_{\beta }^{-}$ from ${\mathrm{\ell }}^{1}\left(ℤ\right)$ to $\mathrm{BV}\left(ℤ\right)$. Here $\mathrm{BV}\left(ℤ\right)$ denotes the set of all functions of bounded variation defined on ℤ. The results we obtained represent significant and natural extensions of what was known previously.

MSC 2010: 42B25; 46E35

This work was supported by the NNSF of China (Grant No. 11701333, 11526122), Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents (Grant No. 2015RCJJ053), Research Award Fund for Outstanding Young Scientists of Shandong Province (Grant No. BS2015SF012) and Support Program for Outstanding Young Scientific and Technological Top-notch Talents of College of Mathematics and Systems Science (Grant No. Sxy2016K01)

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Accepted: 2017-07-07

Published Online: 2018-10-20

Published in Print: 2018-10-25

Citation Information: Mathematica Slovaca, Volume 68, Issue 5, Pages 1097–1112, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918,

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