PreprintCC BY 4.0arXiv
On the equivalence of Sobolev norms in infinite dimensions
Abstract
We prove dimension-free higher-order Sobolev norm estimates on open convex subsets of ℝn with respect to Gaussian measure and use them to obtain norm equivalence on nonempty open convex subsets of ℓ2 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∈[1,∞) and k≥2, including the case p=1, k≥3 left open by Addona--Muratori--Rossi in \cite{AddonaMuratoriRossi}.
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BibTeX
@misc{wang2026equivalence,
title = {{On the equivalence of Sobolev norms in infinite dimensions}},
author = {Zhouzhe Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.24702},
eprint = {2608.24702},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.24702},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the equivalence of Sobolev norms in infinite dimensions
AU - Zhouzhe Wang
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2608.24702
UR - https://arxiv.org/abs/2608.24702
AB - We prove dimension-free higher-order Sobolev norm estimates on open convex subsets of ℝ^n with respect to Gaussian measure and use them to obtain norm equivalence on nonempty open convex subsets of ℓ^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∈[1,∞) and k≥2, including the case p=1, k≥3 left open by Addona--Muratori--Rossi in \cite{AddonaMuratoriRossi}.
ER -