A+ CATEGORY SCIENTIFIC UNIT

Weak-type inequalities for maximal operators acting on Lorentz spaces

Volume 101 / 2014

Adam Osękowski Banach Center Publications 101 (2014), 145-162 MSC: Primary 42B25; Secondary 46E30. DOI: 10.4064/bc101-0-12

Abstract

We prove sharp a priori estimates for the distribution function of the dyadic maximal function $\mathcal{M}\phi$, when $\phi$ belongs to the Lorentz space $L^{p,q}$, $1< p< \infty$, $1\leq q< \infty$. The approach rests on a precise evaluation of the Bellman function corresponding to the problem. As an application, we establish refined weak-type estimates for the dyadic maximal operator: for $p,\,q$ as above and $r\in [1,p]$, we determine the best constant $C_{p,q,r}$ such that for any $\phi\in L^{p,q}$, $$ \|\mathcal{M}\phi\|_{r,\infty}\leq C_{p,q,r}\|\phi\|_{p,q}. $$

Authors

  • Adam OsękowskiDepartment of Mathematics, Informatics and Mechanics
    University of Warsaw
    Banacha 2
    02-097 Warsaw, Poland
    e-mail

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