Characterization of the (fractional) Malliavin-Watanabe-Sobolev spaces $\mathcal{D}^{\alpha,2}$ via the Bargmann-Segal norm

Abstract

Motivated by an open question going back to P.Malliavin and P.-A.Meyer (and closely related to the foundational work of S.Watanabe) on whether Malliavin-Watanabe-Sobolev regularity admits a characterization in terms of a holomorphic Laplace image similar as for Hida distributions, we establish a characterization of the spaces $\mathcal{D}^{\alpha,2}$ for all $\alpha\in\mathbb{R}$ via the Bargmann-Segal norm of the $S$-transform. More precisely, we express $\mathcal{D}^{\alpha,2}$-regularity, $\alpha > 0$, of $F\in L^{2}(\mu)$, as well as dual regularity of distributions, in terms of integrability, differentiability and growth properties of the function \[ (0,1) \ni \lambda \longmapsto \int_{\mathcal{S}'_{\mathbb{C}}} |SF(\lambda u)|^{2}\,d\nu(u) \] involving integer-order derivatives in $\lambda$ for $\alpha\in\mathbb{N}$ and Riemann-Liouville fractional derivatives/integrals for non-integer $\alpha$. Here $\nu$ is the Gaussian Bargmann-Segal measure. This yields practical criteria for both positive and negative (including fractional) orders of Malliavin regularity and thereby bridges Malliavin calculus and Bargmann-Segal techniques from white noise analysis. Applications are worked out for Donsker's delta, self-intersection local times of Gaussian processes, and Gauss kernels.