# On the trace problem for Triebel--Lizorkin spaces with mixed norms

## TL;DR

This paper characterizes the trace spaces of Sobolev spaces with mixed Lebesgue norms on Euclidean space, identifying their structure as mixed-norm Lizorkin--Triebel and Besov spaces, and covers borderline cases and higher order traces.

## Contribution

It provides a comprehensive characterization of trace spaces for mixed-norm Sobolev spaces, including borderline cases and higher order traces, using advanced inequalities and dyadic criteria.

## Key findings

- Trace spaces are mixed-norm Lizorkin--Triebel spaces with specific sum exponents.
- Trace spaces on the last variable are Besov spaces.
- Results include continuous right-inverses and higher order traces.

## Abstract

The subject is traces of Sobolev spaces with mixed Lebesgue norms on Euclidean space. Specifically, restrictions to the hyperplanes given by the first and last coordinates are applied to functions belonging to quasi-homogeneous, mixed-norm Lizorkin--Triebel spaces; Sobolev spaces are obtained from these as special cases. Spaces admitting traces in the distribution sense are characterised except for the borderline cases; these are also covered in case of the first variable. With respect to the first variable the trace spaces are proved to be mixed-norm Lizorkin--Triebel spaces with a specific sum exponent. For the last variable they are similarly defined Besov spaces. The treatment includes continuous right-inverses and higher order traces. The results rely on a sequence version of Nikolskij's inequality, Marschall's inequality for pseudo-differential operators (and Fourier multiplier assertions), as well as dyadic ball criteria.