Weighted Solyanik estimates for the strong maximal function
| Autor: | Paul Hagelstein, Ioannis Parissis |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
maximal function
General Mathematics Dimension (graph theory) Halo function Muckenhoupt weights doubling measure Tauberian conditions 01 natural sciences 42B25 (Primary) 42B35 (Secondary) Combinatorics Doubling measure 0103 physical sciences Classical Analysis and ODEs (math.CA) FOS: Mathematics Maximal operator Locally integrable function 0101 mathematics Reverse holder inequality Mathematics 42B35 010102 general mathematics Mathematical analysis Mathematics - Classical Analysis and ODEs Exponent Maximal function 010307 mathematical physics 42B25 |
| Zdroj: | Recercat: Dipósit de la Recerca de Catalunya Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) Publicacions Matemàtiques; Vol. 62, Núm. 1 (2018); p. 133-159 Recercat. Dipósit de la Recerca de Catalunya instname Publ. Mat. 62, no. 1 (2018), 133-159 Dipòsit Digital de Documents de la UAB Universitat Autònoma de Barcelona |
| ISSN: | 0214-1493 |
| Popis: | Let $\mathsf M_{\mathsf S}$ denote the strong maximal operator on $\mathbb R^n$ and let $w$ be a non-negative, locally integrable function. For $\alpha\in(0,1)$ we define the weighted sharp Tauberian constant $\mathsf C_{\mathsf S}$ associated with $\mathsf M_{\mathsf S}$ by $$ \mathsf C_{\mathsf S} (\alpha):= \sup_{\substack {E\subset \mathbb R^n \\ 0\alpha\}). $$ We show that $\lim_{\alpha\to 1^-} \mathsf C_{\mathsf S} (\alpha)=1$ if and only if $w\in A_\infty ^*$, that is if and only if $w$ is a strong Muckenhoupt weight. This is quantified by the estimate $\mathsf C_{\mathsf S}(\alpha)-1\lesssim_{n} (1-\alpha)^{(cn [w]_{A_\infty ^*})^{-1}}$ as $\alpha\to 1^-$, where $c>0$ is a numerical constant; this estimate is sharp in the sense that the exponent $1/(cn[w]_{A_\infty ^*})$ can not be improved in terms of $[w]_{A_\infty ^*}$. As corollaries, we obtain a sharp reverse H\"older inequality for strong Muckenhoupt weights in $\mathbb R^n$ as well as a quantitative imbedding of $A_\infty^*$ into $A_{p}^*$. We also consider the strong maximal operator on $\mathbb R^n$ associated with the weight $w$ and denoted by $\mathsf M_{\mathsf S} ^w$. In this case the corresponding sharp Tauberian constant $\mathsf C_{\mathsf S} ^w$ is defined by $$ \mathsf C_{\mathsf S} ^w \alpha) := \sup_{\substack {E\subset \mathbb R^n \\ 0\alpha\}).$$ We show that there exists some constant $c_{w,n}>0$ depending only on $w$ and the dimension $n$ such that $\mathsf C_{\mathsf S} ^w (\alpha)-1 \lesssim_{w,n} (1-\alpha)^{c_{w,n}}$ as $\alpha\to 1^-$ whenever $w\in A_\infty ^*$ is a strong Muckenhoupt weight. Comment: 19 pages, submitted for publication |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|