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

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.