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^{{\mathrm{c}}}_{k}} and {M_{k}} , {k\geq 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.