Parabolic equations involving Laguerre operators and weighted mixed-norm estimates

Abstract

<p style='text-indent:20px;'>In this paper, we study evolution equation <inline-formula><tex-math id="M1">\begin{document}$ \partial_t u = -L_\alpha u+f $\end{document}</tex-math></inline-formula> and the corresponding Cauchy problem, where <inline-formula><tex-math id="M2">\begin{document}$ L_\alpha $\end{document}</tex-math></inline-formula> represents the Laguerre operator <inline-formula><tex-math id="M3">\begin{document}$ L_\alpha = \frac 12(-\frac{d^2}{dx^2}+x^2+\frac 1{x^2}(\alpha^2-\frac 14)) $\end{document}</tex-math></inline-formula>, for every <inline-formula><tex-math id="M4">\begin{document}$ \alpha\geq-\frac 12 $\end{document}</tex-math></inline-formula>. We get explicit pointwise formulas for the classical solution and its derivatives by virtue of the parabolic heat-diffusion semigroup <inline-formula><tex-math id="M5">\begin{document}$ \{ e^{-\tau(\partial_t+L_\alpha)}\}_{\tau&gt;0} $\end{document}</tex-math></inline-formula>. In addition, we define the Poisson operator related to the fractional power <inline-formula><tex-math id="M6">\begin{document}$ (\partial_t+L_\alpha)^s $\end{document}</tex-math></inline-formula> and reveal weighted mixed-norm estimates for revelent maximal operators.