Riesz transforms and spectral multipliers of the Hodge-Laguerre operator

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• 作者：Giancarlo Mauceri ; ^{mauceri@dima.unige.it" class="auth_mail" title="E-mail the corresponding author} ; Micol Spinelli ^{micol.spinelli@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：33C50 ; 42C10 ; 42B15 ; 42A50 ; 58A14
• 刊名：Journal of Functional Analysis
• 出版年：2015
• 出版时间：1 December 2015
• 年：2015
• 卷：269
• 期：11
• 页码：3402-3457
• 全文大小：744 K

文摘

On ${\mathbb{R}}_{+}^d$, endowed with the Laguerre probability measure 渭_伪${渭}_{伪}$, we define a Hodge–Laguerre operator L_伪=未未^{鈦?/sup>+未鈦?/sup>未}${\mathbb{L}}_{伪}=未{未}^{鈦?/mo>}+{未}^{鈦?/mo>}未$ acting on differential forms. Here 未   is the Laguerre exterior differentiation operator, defined as the classical exterior differential, except that the partial derivatives ∂_xi${\partial }_{x_i}$ are replaced by the ‘‘Laguerre derivatives’’ $\sqrt{x_i}{\partial }_{x_i}$, and 未^鈦?/sup>${未}^{鈦?/mo>}$ is the adjoint of 未   with respect to inner product on forms defined by the Euclidean structure and the Laguerre measure 渭_伪${渭}_{伪}$. We prove dimension-free bounds on L^p$L^p$, 1<p<∞$1<p<\infty$, for the Riesz transforms $未{\mathbb{L}}_{伪}^{-1/2}$ and ${未}^{鈦?/mo>}{\mathbb{L}}_{伪}^{-1/2}$. As applications we prove the strong Hodge–de Rahm–Kodaira decomposition for forms in L^p$L^p$ and deduce existence and regularity results for the solutions of the Hodge and de Rham equations in L^p$L^p$. We also prove that for suitable functions m   the operator m(L_伪)$m({\mathbb{L}}_{伪})$ is bounded on L^p$L^p$, 1<p<∞$1<p<\infty$.