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

Abstract

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 , s c {P_{k,{\mathrm{s}}}^{{\mathrm{c}}}} , P k , s {P_{k,{\mathrm{s}}}} , P k , w c {P_{k,{\mathrm{w}}}^{{\mathrm{c}}}} and P k , w {P_{k,{\mathrm{w}}}} , the sets of all p ∈ [ 1 , ∞ ] {p\in[1,\infty]} for which the weak and strong type ( p , p ) {(p,p)} inequalities hold for the centered and non-centered modified Hardy–Littlewood maximal operators M k c {M^{{\mathrm{c}}}_{k}} and M k {M_{k}} , k ≥ 1 {k\geq 1} . For any fixed k we describe the necessary conditions that P k , s c {P_{k,{\mathrm{s}}}^{{\mathrm{c}}}} , P k , s {P_{k,{\mathrm{s}}}} , P k , w c {P_{k,{\mathrm{w}}}^{{\mathrm{c}}}} and P k , w {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.