Products and Convolutions in Hermite-Sobolev spaces

2026/07/31 by Suprio Bhar, Rajeev Bhaskaran
Mathematics · #math.FA #msc:46A11 #msc:46F10 #msc:46H99 #msc:35A23 #msc:46C05 #msc:47J05

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arxiv created 2026/07/31 · arxiv updated 2026/08/03

Abstract

In this paper, we show that the product, or equivalently the convolutions of two functions in the Hermite-Sobolev spaces S_p(ℝ^d) is again in the same space, for p depending on the dimension d. As a consequence, for such p we show that the product, or equivalently the convolutions of ϕ∈ S_p(ℝ^d) and ψ∈ S_-p(ℝ^d) is in S_-p(ℝ^d). As a further consequence, we show that the operators of translation by x on S_p(ℝ^d) are bounded uniformly in x ∈ ℝ^d.

Citations

• Existence and Uniqueness of Stochastic PDEs associated with the Forward Equations: An Approach using Alternate Norms
• Probabilistic Representations of Solutions of the Forward Equations
• Probabilistic representations of solutions to the heat equation