On relations between weak and strong type inequalities for modified maximal operators on non-doubling metric measure spaces

Abstract

In this article we investigate a special class of non-doubling metric measure spaces in order to describe the possible configurations of $P_{k,\mathrm{s}}^{\mathrm{c}}$, $P_{k,\mathrm{s}}$, $P_{k,\mathrm{w}}^{\mathrm{c}}$ and $P_{k,\mathrm{w}}$, the sets of all $p\in [1,\infty ]$ for which the weak and strong type $(p,p)$ inequalities hold for the centered and non-centered modified Hardy--Littlewood maximal operators, $M_k^{\mathrm{c}}$ and $M_k$, $k\ge 1$. For any fixed $k$ we describe the necessary conditions that $P_{k,\mathrm{s}}^{\mathrm{c}}$, $P_{k,\mathrm{s}}$, $P_{k,\mathrm{w}}^{\mathrm{c}}$ and $P_{k,\mathrm{w}}$ must satisfy in general and illustrate each admissible configuration with a properly chosen non-doubling metric measure space. We also give some partial results related to an analogous problem stated for varying $k$.