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 $p \in (1,\infty)$ and a metric measure space $\mathcal X = (X, \rho, \mu)$ we let $\Omega^p_{\rm HL}(\mathcal X) \subset [0,1]^2$ be the set of all pairs $(\frac{1}{q},\frac{1...

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Bibliographic Details
Author: Kosz, D.
Format: article
Status:Published version
Publication Date:2023
Country:España
Institution:Basque Center for Applied Mathematics (BCAM)
Repository:BIRD. BCAM's Institutional Repository Data
OAI Identifier:oai:bird.bcamath.org:20.500.11824/1727
Online Access:http://hdl.handle.net/20.500.11824/1727
Access Level:Open access
Keyword:centered Hardy–Littlewood maximal operator
Lorentz space
nondoubling metric measure space
Description
Summary: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 $\mathcal X = (X, \rho, \mu)$ we let $\Omega^p_{\rm HL}(\mathcal X) \subset [0,1]^2$ be the set of all pairs $(\frac{1}{q},\frac{1}{r})$ such that $\mathcal M$ is bounded from $L^{p,q}(\mathcal X)$ to $L^{p,r}(\mathcal X)$. Under mild assumptions on $\mu$, for each fixed $p$ all possible shapes of $\Omega^p_{\rm HL}(\mathcal X)$ are characterized. Namely, we show that the boundary of $\Omega^p_{\rm HL}(\mathcal X)$ either is empty or takes the form $\{ \delta \} \times [0, \lim_{u \rightarrow \delta} F(u)] \cup \{(u, F(u)) : u \in (\delta, 1] \}$, where $\delta \in [0,1]$ and $F \colon [\delta, 1] \rightarrow [0,1]$ is concave, nondecreasing, and satisfies $F(u) \leq u$. Conversely, for each such $F$ we find $\mathcal X$ such that $\mathcal M$ is bounded from $L^{p,q}(\mathcal X)$ to $L^{p,r}(\mathcal X)$ if and only if the point $(\frac{1}{q}, \frac{1}{r})$ lies on or under the graph of $F$, that is, $\frac{1}{q} \geq \delta$ and $\frac{1}{r} \leq F\big(\frac{1}{q}\big)$.

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