Parabolic equations involving Laguerre operators and weighted mixed-norm estimates

Huiying Fan; Tao Ma

doi:10.3934/cpaa.2020249

Article Contents

2020, Volume 19, Issue 12: 5487-5508. Doi: 10.3934/cpaa.2020249

This issue Previous Article Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry Next Article A truncated real interpolation method and characterizations of screened Sobolev spaces

Parabolic equations involving Laguerre operators and weighted mixed-norm estimates

Huiying Fan^1, , and
Tao Ma^2,

1.
School of Mathematical Science, Zhejiang University, Hangzhou 310027, China
2.
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

^* Corresponding author
^* Corresponding author

Received: February 29, 2020

Revised: May 31, 2020

Early access: October 2020

Published: December 2020

The second author was supported by National Natural Science Foundation of China (Grant Nos. 11671308, 11971431)

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In this paper, we study evolution equation $ \partial_t u = -L_\alpha u+f $ and the corresponding Cauchy problem, where $ L_\alpha $ represents the Laguerre operator $ L_\alpha = \frac 12(-\frac{d^2}{dx^2}+x^2+\frac 1{x^2}(\alpha^2-\frac 14)) $, for every $ \alpha\geq-\frac 12 $. We get explicit pointwise formulas for the classical solution and its derivatives by virtue of the parabolic heat-diffusion semigroup $ \{ e^{-\tau(\partial_t+L_\alpha)}\}_{\tau>0} $. In addition, we define the Poisson operator related to the fractional power $ (\partial_t+L_\alpha)^s $ and reveal weighted mixed-norm estimates for revelent maximal operators.

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Mathematics Subject Classification: Primary: 35C15, 47D03; Secondary: 42B25.

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