Traces of Sobolev functions on regular surfaces in infinite dimensions

TL;DR

This paper develops a theory for defining and analyzing the traces of Sobolev functions on regular surfaces within infinite-dimensional Banach spaces equipped with Gaussian measures, extending classical boundary trace concepts.

Contribution

It introduces a framework for trace operators of Sobolev functions on surfaces in infinite dimensions and characterizes their range, including an integration by parts formula involving these traces.

Findings

Trace operators map Sobolev functions to L^q spaces on the boundary surface.

Range of the trace operator is contained in L^q spaces, with conditions for L^p inclusion.

In special cases, the trace space is characterized as a fractional Sobolev space.

Abstract

In a Banach space $X$ endowed with a nondegenerate Gaussian measure, we consider Sobolev spaces of real functions defined in a sublevel set $O= \{x\in X:\;G(x) <0\}$ of a Sobolev nondegenerate function $G:X\mapsto \R$. We define the traces at $G^{-1}(0)$ of the elements of $W^{1,p}(O, \mu)$ for $p>1$, as elements of $L^1(G^{-1}(0), \rho)$ where $\rho$ is the surface measure of Feyel and de La Pradelle. The range of the trace operator is contained in $L^q(G^{-1}(0), \rho)$ for $1\leq q<p$ and even in $L^p(G^{-1}(0), \rho)$ under further assumptions. If $O$ is a suitable halfspace, the range is characterized as a sort of fractional Sobolev space at the boundary. An important consequence of the general theory is an integration by parts formula for Sobolev functions, which involves their traces at $G^{-1}(0)$.