On the trace embedding and its applications to evolution equations

TL;DR

This paper develops a unified theory for trace embeddings at initial times for functions with mixed smoothness, applicable to evolution equations, including fractional and stochastic types, with broad interpolation capabilities.

Contribution

It extends existing trace embedding results to general interpolation couples, unifying various previous findings and applying to regularity problems in evolution equations.

Findings

Unified trace embedding results for mixed smoothness functions.

Application to regularity in fractional evolution equations.

Uniform trace estimates for stochastic evolution equations.

Abstract

In this paper we consider traces at initial times for functions with mixed time-space smoothness. Such results are often needed in the theory of evolution equations. Our result extends and unifies many previous results. Our main improvement is that we can allow general interpolation couples. The abstract results are applied to regularity problems for fractional evolution equations and stochastic evolution equation, where uniform trace estimates on the half-line are shown.