Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
134 tokens/sec
GPT-4o
9 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Regularity and continuity of the multilinear strong maximal operators (1801.09828v2)

Published 30 Jan 2018 in math.CA and math.AP

Abstract: Let $m\ge 1$, in this paper, our object of investigation is the regularity and and continuity properties of the following multilinear strong maximal operator $${\mathscr{M}}{\mathcal{R}}(\vec{f})(x)=\sup{\substack{R \ni x R\in\mathcal{R}}}\prod\limits_{i=1}m\frac{1}{|R|}\int_{R}|f_i(y)|dy,$$ where $x\in\mathbb{R}d$ and $\mathcal{R}$ denotes the family of all rectangles in $\mathbb{R}d$ with sides parallel to the axes. When $m=1$, denote $\mathscr{M}{\mathcal{R}}$ by $\mathcal {M}{\mathcal{R}}$.Then, $\mathcal {M}{\mathcal{R}}$ coincides with the classical strong maximal function initially studied by Jessen, Marcinkiewicz and Zygmund. We showed that ${\mathscr{M}}{\mathcal{R}}$ is bounded and continuous from the Sobolev spaces $W{1,p_1}(\mathbb{R}d)\times \cdots\times W{1,p_m}(\mathbb{R}d)$ to $W{1,p} (\mathbb{R}d)$, from the Besov spaces $B_{s}{p_1,q} (\mathbb{R}d)\times\cdots\times B_s{p_m,q}(\mathbb{R}d)$ to $B_s{p,q}(\mathbb{R}d)$, from the Triebel-Lizorkin spaces $F_{s}{p_1,q}(\mathbb{R}d)\times\cdots\times F_s{p_m,q}(\mathbb{R}d)$ to $F_s{p,q}(\mathbb{R}d)$. As a consequence, we further showed that ${\mathscr{M}}{\mathcal{R}}$ is bounded and continuous from the fractional Sobolev spaces $W{s,p_1}(\mathbb{R}d)\times \cdots\times W{s,p_m}(\mathbb{R}d)$ to $W{s,p}(\mathbb{R}d)$ for $0<s\leq 1$ and $1<p<\infty$. As an application, we obtain a weak type inequality for the Sobolev capacity, which can be used to prove the $p$-quasicontinuity of $\mathscr{M}{\mathcal{R}}$. The discrete type of the strong maximal operators has also been considered. We showed that this discrete type of the maximal operators enjoys somewhat unexpected regularity properties.

Summary

We haven't generated a summary for this paper yet.