Boundedness properties of maximal operators on Lorentz spaces

Dariusz Kosz¹

¹Basque Center for Applied Mathematics, and Wrocław University of Science and Technology

Annales Fennici Mathematici, Tome 48 (2023) no. 2, pp. 515-535

Résumé

We study mapping properties of the centered Hardy-Littlewood maximal operator $M$ acting on Lorentz spaces. Given $p \in (1,\infty)$ and a metric measure space $X = (X, \rho, \mu)$ we let $\Omega^p_{\rm HL}(X) \subset [0,1]^2$ be the set of all pairs $(\frac{1}{q},\frac{1}{r})$ such that $M$ is bounded from $L^{p,q}(X)$ to $L^{p,r}(X)$. Under mild assumptions on $\mu$, for each fixed $p$ all possible shapes of $\Omega^p_{\rm HL}(X)$ are characterized. Namely, we show that the boundary of $\Omega^p_{\rm HL}(X)$ either is empty or takes the form $\{ \delta \} \times [0, \lim_{u \rightarrow \delta} F(u)] \cup \{(u, F(u)) \colon u \in (\delta, 1] \}$, 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 $X$ such that $M$ is bounded from $L^{p,q}(X)$ to $L^{p,r}(X)$ 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 $\frac{1}{r} \leq F\big(\frac{1}{q}\big)$.

Détail

DOI : 10.54330/afm.131758

Keywords: Centered Hardy-Littlewood maximal operator, Lorentz space, nondoubling metric measure space