"$H=W$" in infinite dimensions

TL;DR

This paper extends the classical finite-dimensional $H^{m,p}=W^{m,p}$ Sobolev identity to infinite-dimensional settings by proving that smooth cylinder functions are dense in $W^{m,p}(O)$ for open subsets $O$ of $\ell^2$ with smooth boundary. The authors develop an infinite-dimensional analogue of truncation, boundary-straightening, and partition of unity to establish the density, which yields $H^{m,p}(O)=W^{m,p}(O)$. A key technical achievement is the compact truncation step, which also yields sharp Schatten-$p$-norm type estimates for higher-order derivatives of the Gross convolution in the $p=2$ case. The results unify various infinite-dimensional Sobolev definitions (Bogachev’s frameworks) and provide a robust method for extending finite-dimensional PDE and Malliavin-calculus techniques to infinite-dimensional domains.

Abstract

It is well known that $H^{m,p}(Ω) = W^{m,p}(Ω)$ holds for any $m, n \in \mathbb{N}$, $p \in [1, \infty)$, and open subset $Ω$ of $\mathbb{R}^n$. Due to the essential difficulty that there exists no nontrivial translation-invariant measure in infinite dimensions, it is hard to obtain its infinite-dimensional counterparts. In this paper, using infinite-dimensional analogues of the classical techniques of truncation, boundary straightening, and partition of unity, we prove that smooth cylinder functions are dense in $W^{m,p}(O)$. Consequently, $H^{m,p}(O) = W^{m,p}(O)$ holds for any $m \in \mathbb{N}$, $p \in [1, \infty)$, and open subset $O$ of $\ell^2$ with a suitable boundary. Moreover, in the key step of compact truncation, we also prove that the Schatten $p$-norm type estimates for the higher-order derivatives of the Gross convolution are sharp for $p = 2$.

Theorems & Definitions (67)

• Definition 1
• Definition 2
• Lemma 3
• Lemma 4
• proof
• Lemma 5
• proof
• Remark 6
• Definition 7
• Remark 8