"$H=W$" in infinite dimensions

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 $p = 2$.

💡 Research Summary

The paper tackles the long‑standing problem of extending the classical Sobolev equivalence “H = W” to infinite‑dimensional settings. In finite dimensions the equality of the Bessel potential space H^{m,p}(Ω) and the Sobolev space W^{m,p}(Ω) follows from the existence of a bounded extension operator and from standard tools such as truncation, mollification, and partition of unity. In an infinite‑dimensional Hilbert space ℓ² equipped with a Gaussian measure P, these tools are unavailable because there is no non‑trivial translation‑invariant measure. The authors therefore develop infinite‑dimensional analogues of the three classical techniques and prove that smooth cylinder functions are dense in the Sobolev space W^{m,p}(O) for any open set O⊂ℓ² with a sufficiently smooth boundary. Consequently the closure of smooth cylinder functions, denoted H^{m,p}(O), coincides with W^{m,p}(O) for every integer m≥1 and every p∈