Reduction of integration domain in Triebel-Lizorkin spaces

TL;DR

This paper studies how truncating the integration domain based on boundary distance affects the equivalence of Triebel-Lizorkin and Sobolev seminorms, providing examples where comparability holds or fails.

Contribution

It offers a detailed analysis of the conditions under which seminorms remain comparable when the domain is truncated, including various kernels and domain types.

Findings

Comparability holds for certain kernels and domains.

Examples where comparability does not hold.

Guidelines for when truncation preserves seminorm equivalence.

Abstract

We investigate the comparability of generalized Triebel--Lizorkin and Sobolev seminorms on uniform and non-uniform sets when the integration domain is truncated according to the distance from the boundary. We provide numerous examples of kernels and domains in which the comparability does and does not hold.