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Boundedness properties of maximal operators on Lorentz spaces in non-doubling setting

Research paper by Dariusz Kosz

Indexed on: 11 Oct '19Published on: 08 May '19Published in: arXiv - Mathematics - Classical Analysis and ODEs



Abstract

We study mapping properties of the centered Hardy--Littlewood maximal operator $\mathcal{M}$ acting on Lorentz spaces $L^{p,q}(\mathfrak{X})$ in the context of certain non-doubling metric measure spaces $\mathfrak{X}$. The special class of spaces for which these properties are very peculiar is considered. In particular, for fixed $p \in (1, \infty)$, $\delta \in [0,1)$ and any concave, non-decreasing function $F \colon [\delta, 1] \rightarrow [0,1]$ satisfying $F(u) \leq u$, $u \in [\delta, 1]$, we construct a space $\mathfrak{X}$ for which the associated operator $\mathcal{M}$ is bounded from $L^{p,q}(\mathfrak{X})$ to $L^{p,r}(\mathfrak{X})$ if and only if the point $(\frac{1}{q}, \frac{1}{r}) \in [0,1]^2$ lies under the graph of $F$, that is $\frac{1}{q} \geq \delta$ and $\frac{1}{r} \leq F\big(\frac{1}{q}\big)$. The analogous result for functions whose domains are of the form $(\delta, 1]$ is also given.