Boundedness properties of maximal operators on Lorentz spaces

We study mapping properties of the centered Hardy–Littlewood maximal operator $\mathcal{M}$ acting on Lorentz spaces. Given $p\in (1,\mathrm{\infty })$ and a metric measure space $\mathfrak{X}$ we let $\Omega^p_{\rm HL}(\mathfrak{X}) \subset [0,1]^2$bethesetofallpairs$(\frac{1}{q},\frac{1}{r})$suchthat$$\mathcal{M}$$isboundedfrom$L^{p,q}(\mathfrak{X})$to$L^{p,r}(\mathfrak{X})$\mathrm{.}Foreachfixed$p$allpossibleshapesof$\Omega^p_{\rm HL}(\mathfrak{X})$ are characterized. Namely, we show that the boundary of ${\mathrm{\Omega }}_{\mathrm{H}\mathrm{L}}^p(\mathfrak{X})$ either is empty or takes the form $$\{\delta \}\times [0,\underset{u\to \delta }{\lim }F(u)]\cup \{(u,F(u)):u\in (\delta ,1]\},$$ where $\delta \in [0,1]$ and $F \colon [\delta, 1] \rightarrow [0,1]$isconcave,non-decreasing,andsatisfying$F(u) \leq u$. Conversely, for each such $F$ we find $\mathfrak{X}$ such that $\mathcal{M}$ is bounded from $L^{p,q}(\mathfrak{X})$ to $L^{p,r}(\mathfrak{X})$ if and only if the point $(\frac{1}{q},\frac{1}{r})$ lies on or under the graph of $F$, that is, $\frac{1}{q}\ge \delta$ and $\frac{1}{r}\le F\left(\frac{1}{q}\right)$.