arXiv subjects

Zhouzhe Wang

Publications and source records attributed to Zhouzhe Wang.

6 recordsLinked to original sources

arXiv2026

On the equivalence of Sobolev norms in infinite dimensions

We prove dimension-free higher-order Sobolev norm estimates on open convex subsets of $\mathbb{R}^n$ with respect to Gaussian measure and use them to obtain norm equivalence on nonempty open convex subsets of $\ell^2$ endowed with a nondegenerate Gaussian measure. To the best of our knowledge, this is the first such equivalence theorem on a proper open subset of an infinite-dimensional Hilbert space, beyond the earlier whole-space results. We also prove the Malliavin--Sobolev norm equivalence for all $p\in[1,\infty)$ and $k\ge2$, including the case $p=1$, $k\ge3$ left open by Addona--Muratori--Rossi in \cite{AddonaMuratoriRossi}.

math.FA↗

arXiv2026

Measures on General Codimensional Surfaces in Infinite Dimensions and Stokes-Type Theorems

In this paper, we give an explicit construction of surface measures on a class of surfaces with arbitrary, possibly infinite, codimension in $\ell^2$. These measures are constructed from local representations associated with a fixed Gaussian product measure. We then establish local and global Gauss--Green-type formulas, introduce a notion of top-degree differential form, and derive an associated Stokes-type identity. We also determine the orientation of the boundary induced by the orientation of the surface. Moreover, therelationship between $\mathcal F$-continuity and Borel measurability is examined,which reveals a phenomenon specific to the infinite-dimensional setting.

math.FA↗

arXiv2026

Extension of Sobolev functions on balls in infinite dimensions

We prove the existence of a bounded Sobolev extension operator $E:W^{p,1}\left( B,P \right) \rightarrow W^{p,1}\left( \ell^{2} ,P \right)$ using a completely new method, where $B\subset \ell^{2}$ is the unit ball and $P$ is any non-trivial centered Gaussian measure on $\ell^{2}$. This solves an open problem posed in the literatures.

math.FA↗

arXiv2026

"$H=W$" in infinite dimensions

The classical ``$H=W$" theorem establishes the identity between two function spaces on an arbitrary nonempty open set in the Euclidean spaces: the space $W$ defined via weak derivatives, and the space $H$ defined as the closure of smooth functions within $W$ space. Extending this result to infinite-dimensional spaces is challenging due to the lack of a nontrivial translation-invariant measure and the proliferation of infinite sums inherent to infinite dimensions. In this paper, by adapting several techniques developed in our previous works, we prove that smooth functions are dense in the Sobolev space of functions on arbitrary non-empty open set in $\ell^2$, thereby establishing an infinite-dimensional counterpart of ``$H=W$". Such density results reduce the problem of deriving a priori $L^2$ estimates for differential operators -- originating from the classical Fredholm alternative and Carleman estimates -- to the simpler case of smooth functions. If approximation by smooth cylindrical functions is possible, the problem can be reduced to calculus. Unfortunately, this does not hold for every open set in $\ell^2$. However, we prove that such an approximation does hold on open sets that satisfy the segment condition.

math.FA↗

arXiv2024

Smooth plurisubharmonic exhaustion functions on pseudo-convex domains in infinite dimensions

In this paper, by modifying significantly the Friedrichs-Gross mollifier technique and/or using the Lasry-Lions regularization technique together with some carefully chosen cut-off functions, for the first time we construct explicitly smooth exhaustion functions on any open subset and smooth plurisubharmonic exhaustion functions on any pseudo-convex domain in a typical Hilbert space, which enjoy delicate properties needed for the $L^{2}$ method in infinite-dimensional complex analysis.

math.CV↗

arXiv2024

$L^2$ estimates and existence theorems for the $\overline{\partial}$ operators in infinite dimensions, II

This paper is the second part of our series of works to establish $L^2$ estimates and existence theorems for the $\overline{\partial}$ operators in infinite dimensions. In this part, we consider the most difficult case, i.e., the underlying space is a general pseudo-convex domain. In order to solve this longstanding open problem, we introduce several new concepts and techniques, which have independent interest and pave the way for research that investigates some other issues in infinite-dimensional analysis.

math.FA↗