# Comparison of spaces of Hardy type for the Ornstein-Uhlenbeck operator - Mathematics > Functional Analysis

Comparison of spaces of Hardy type for the Ornstein-Uhlenbeck operator - Mathematics > Functional Analysis - Download this document for free, or read online. Document in PDF available to download.

Abstract: Denote by g the Gauss measure on R^n and by L the Ornstein-Uhlenbeckoperator. In this paper we introduce a local Hardy space h^1g of Goldbergtype and we compare it with the Hardy space H^1g introduced in a previouspaper by Mauceri and Meda. We show that for each each positive r the imaginarypowers of the operator rI+L are unbounded from h^1g to L^1g. This result isin sharp contrast both with the fact that the imaginary powers are bounded from$H^1g}$ to L^1g, and with the fact that for the Euclidean laplacian \Deltaand the Lebesgue measure \lambda the imaginary powers of rI-\Delta are boundedfrom the Goldberg space h^1\lambda to L^1\lambda. We consider also the caseof Riemannian manifolds M with Riemannian measure m. We prove that, undercertain geometric assumptions on M, an operator T, bounded on L^2m, and witha kernel satisfying certain analytic assumptions, is bounded from H^1m toL^1m if and only if it is bounded from h^1m to L^1m. Here H^1m denotesthe Hardy space on locally doubling metric measure spaces introduced by theauthors in arXiv:0808.0146, and h^1m is a Goldberg type Hardy space on M,equivalent to a space recently introduced by M. Taylor. The case of translationinvariant operators on homogeneous trees is also considered.

Author: ** A. Carbonaro, G. Mauceri, S. Meda**

Source: https://arxiv.org/