Riesz Transforms and Spectral Multipliers of the Hodge-Laguerre Operator

Abstract

On $\mathbb{R}^d_+$, endowed with the Laguerre probability measure $\mu_\alpha$, we define a Hodge-Laguerre operator $\mathbb{L}_\alpha=\delta\delta^*+\delta^* \delta$ acting on differential forms. Here $\delta$ 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 $\delta^*$ is the adjoint of $\delta$ with respect to inner product on forms defined by the Euclidean structure and the Laguerre measure $\mu_\alpha$. We prove dimension-free bounds on $L^p$, $1<p<\infty$, for the Riesz transforms $\delta \mathbb{L}_\alpha^{-1/2}$ and $\delta^* \mathbb{L}_ \alpha^{-1/2}$. 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 $m(\mathbb{L}^\alpha)$ is bounded on $L^p$, $1<p<\infty$.