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Traces of Sobolev functions on regular surfaces in infinite dimensions (1302.2204v1)

Published 9 Feb 2013 in math.AP

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 $L1(G{-1}(0), \rho)$ where $\rho$ is the surface measure of Feyel and de La Pradelle. The range of the trace operator is contained in $Lq(G{-1}(0), \rho)$ for $1\leq q<p$ and even in $Lp(G{-1}(0), \rho)$ 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 $G{-1}(0)$.

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