Riesz Transforms and Spectral Multipliers of the Hodge-Laguerre Operator

Abstract

On ${\mathbb{R}}_{+}^d$, endowed with the Laguerre probability measure $\hat{I}{¼}_{\hat{I}}\pm$, we define a Hodge-Laguerre operator ${\mathbb{L}}_{\hat{I}}\pm =\hat{I}´\hat{I}{´}^{\ast }+\hat{I}{´}^{\ast }\hat{I}´$ acting on differential forms. Here $\hat{I}´$ is the Laguerre exterior differentiation operator, defined as the classical exterior differential, except that the partial derivatives ${\partial }_{x_i}$ are replaced by the "Laguerre derivatives" $\sqrt{x_i}{\partial }_{x_i}$, and $\hat{I}{´}^{\ast }$ is the adjoint of $\hat{I}´$ with respect to inner product on forms defined by the Euclidean structure and the Laguerre measure $\hat{I}{¼}_{\hat{I}}\pm$. We prove dimension-free bounds on $L^p$, $1<p<\infty$, for the Riesz transforms $\hat{I}´{\mathbb{L}}_{\hat{I}}{\pm }^{-1/2}$ and $\hat{I}{´}^{\ast }{\mathbb{L}}_{\hat{I}}{\pm }^{-1/2}$. As applications we prove the strong Hodge-de Rahm-Kodaira decomposition for forms in $L^p$ and deduce existence and regularity results for the solutions of the Hodge and de Rham equations in $L^p$. We also prove that for suitable functions $m$ the operator $m({\mathbb{L}}^{\hat{I}}\pm )$ is bounded on $L^p$, $1<p<\infty$.

49 pages