Generalized BMO-type seminorms and vector-valued Sobolev functions

TL;DR

This paper introduces a unified framework of BMO-type seminorms to characterize Sobolev and related spaces, providing new pointwise limit theorems and non-distributional characterizations for scalar and vector-valued functions.

Contribution

It develops novel BMO-type seminorms that extend existing results and characterize Sobolev spaces without distributional derivatives, including vector-valued cases.

Findings

Seminorms converge to integral functionals with convex, p-homogeneous integrands.

Provides non-distributional characterizations of Sobolev spaces.

Establishes a pointwise limit theorem for parameter-dependent BMO-type seminorms.

Abstract

We establish a pointwise limit theorem for a broad class of pa\-ra\-me\-ter-\-de\-pen\-dent BMO-type seminorms as the parameter tends to zero. By introducing novel BMO-type seminorms, we provide a unified framework that extends several existing results and yields non-distributional characterizations of Sobolev-type spaces, both in the scalar and in the vector-valued setting. More precisely, for any open set $\Omega\subset \mathbb{R}^n$ and any $p\in (1, \infty)$, we provide a characterization of the Sobolev space $W^{1,p}(\Omega; \mathbb{R}^m)$. In addition, we characterize the space $E^{1,p}(\Omega;\mathbb{R}^n)$ of $L^p$ maps with $p$-integrable distributional symmetric gradient.\\ Finally, for all $p\in [1, \infty)$, we show that these seminorms converge to integral functionals with convex, $p$-homogeneous integrands associated with the distributional gradient and the symmetric gradient.