Regularity and continuity of local Multilinear Maximal type operator

TL;DR

This paper investigates the regularity and continuity of local multilinear fractional maximal operators, providing new pointwise derivative estimates and establishing their boundedness in Sobolev spaces.

Contribution

It introduces novel pointwise estimates for derivatives of local multilinear maximal functions, enabling new Sobolev space norm inequalities and boundary value bounds.

Findings

New pointwise estimates for derivatives of maximal functions

Boundedness results in Sobolev spaces

Bounds for operators with zero boundary values

Abstract

This paper will be devoted to study the regularity and continuity properties of the following local multilinear fractional type maximal operators, $$\mathfrak{M}_{\alpha,\Omega}(\vec{f})(x)=\sup\limits_{0<r<{\rm dist}(x,\Omega^c)}\frac{r^\alpha}{|B(x,r)|^m}\prod\limits_{i=1}^m\int_{B(x,r)}|f_i(y)|dy,\quad \hbox{for \ }0\leq\alpha<mn,$$ where $\Omega$ is a subdomain in $\mathbb{R}^n$, $\Omega^c=\mathbb{R}^n\backslash\Omega$ and $B(x,r)$ is the ball in $\mathbb{R}^n$ centered at $x$ with radius $r$. Several new pointwise estimates for the derivative of the local multilinear maximal function $\mathfrak{M}_{0,\Omega}$ and the fractional maximal functions $\mathfrak{M}_{\alpha,\Omega}$ $(0<\alpha< mn)$ will be presented. These estimates will not only enable us to establish certain norm inequalities for these operators in Sobolev spaces, but also give us the opportunity to obtain the bounds of these operators on the Sobolev space with zero boundary values.