Traces of Sobolev functions on regular surfaces in infinite dimensions

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∈X:G(x)<0} of a Sobolev nondegenerate function G:X↦R. We define the traces at G−1(0) of the elements of W1,p(O,μ) for p>1, as elements of L1(G−1(0),ρ) where ρ is the surface measure of Feyel and de La Pradelle. The range of the trace operator is contained in Lq(G−1(0),ρ) for 1⩽q<p and even in Lp(G−1(0),ρ) 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).