Bargmann transfoms associated with reproducing kernel Hilbert space and application to Dirichlet spaces

TL;DR

This paper develops new Bargmann-type integral transforms for reproducing kernel Hilbert spaces, including the Dirichlet space, providing explicit isometries, characterizations, and spectral analysis, extending classical results.

Contribution

It introduces explicit Bargmann-type isometries for RKHSs, especially the Dirichlet space, and characterizes these spaces as harmonic spaces of elliptic PDEs.

Findings

Constructed a Bargmann-type integral isometry for RKHSs.

Derived an explicit inverse isometry for certain measure spaces.

Provided a new spectral characterization of the Dirichlet space.

Abstract

The aim of the present paper is three folds. For a reproducing kernel Hilbert space $\mathcal{A}$ (R.K.H.S) and a $\sigma-$finite measure space $(M_{1},d\mu_{1})$ for which the corresponding $L^{2}-$space is a separable Hilbert space, we first build an isometry of Bargmann type as an integral transform from $L^{2}(M_{1},d\mu_{1})$ into $\mathcal{A}$. Secondly, in the case where there exists a $\sigma-$finite measure space $(M_{2},d\mu_{2})$ such that the Hilbert space $L^{2}(M_{2},d\mu_{2})$ is separable and $\mathcal{A}\subset L^{2}(M_{2},d\mu_{2})$ the inverse isometry is also given in an explicit form as an integral transform. As consequence, we recover some classical isometries of Bargmann type. Thirdly, for the classical Dirichlet space as R.K.H.S, we elaborate a new isometry of Bargmann type. Furthermore, for this Dirichlet space, we give a new characterization, as harmonic space of a single second order elliptic partial differential operator for which, we present some spectral properties. Finally, we extend the same results to a class of generalized Bergman-Dirichlet space.