Boundedness properties of maximal operators on Lorentz spaces

Abstract

We study mapping properties of the centered Hardy--Littlewood maximal operator $\mathcal{M}$ acting on Lorentz spaces. Given $p\in (1,\infty )$ and a metric measure space $\mathfrak{X}$ we let ${\Omega }_{\mathrm{HL}}^p(\mathfrak{X})\subset [0,1{]}^2$ be the set of all pairs $(\frac{1}{q},\frac{1}{r})$ such that $\mathcal{M}$ is bounded from $L^{p,q}(\mathfrak{X})$ to $L^{p,r}(\mathfrak{X})$. For each fixed $p$ all possible shapes of ${\Omega }_{\mathrm{HL}}^p(\mathfrak{X})$ are characterized. Namely, we show that the boundary of ${\Omega }_{\mathrm{HL}}^p(\mathfrak{X})$ 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 $F:[\delta ,1]\to [0,1]$ is concave, non-decreasing, and satisfying $F(u)\le u$. 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 $\frac{1}{r}\le F\left(\frac{1}{q}\right)$.

arXiv comment20 pages, 2 figures