Non-centered maximal operators over ${\ell }^q$ balls for exponential and Laguerre-type measures

Abstract

We establish the \(L^p\)-boundedness, \(1<p<\infty\), of the non-centered Hardy--Littlewood maximal operator associated with \(\ell^q\) balls, \(1<q<\infty\), for the non-radial exponential measure \[ dμ(x)=e^{-|x_1|-\cdots-|x_d|}\,dx \] in every dimension \(d\ge1\). This settles the higher-dimensional Euclidean case left open by Nowak, Sasso, Sjögren and Stempak, and extends their results to the full range \(1<q<\infty\). We also prove that these operators are not of weak type \((1,1)\) when \(d\ge2\). Together with the known results for \(q=1\) and \(q=\infty\) for the exponential measure, this gives a complete description of the \(L^p\)-boundedness and weak type \((1,1)\) behavior for all \(1\le q\le\infty\). {We also extend the strong \(L^p\)-boundedness to the Laguerre-type measures \(dμ_α(x)=\prod_{j=1}^d|x_j|^{α_j}e^{-|x_1|-\cdots-|x_d|}\,dx\) for all \(α_j\ge0\) and \(1\le q\le\infty\), and for the full range \(α_j>-1\) when \(q=\infty\).} {The main arguments rely} on geometric estimates for sections of \(\ell^q\) balls and a slicing argument.

arXiv comment24 pages