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

Research paper by **Dariusz Kosz**

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

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#### 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.