Hardy inequalities for the fractional powers of the Grushin operator

Manli Song; Jinggang Tan

doi:10.3934/cpaa.2020192

Article Contents

2020, Volume 19, Issue 9: 4699-4726. Doi: 10.3934/cpaa.2020192

This issue Previous Article Degenerate coercive quasilinear elliptic equations with subcritical or critical exponents in

$ \mathbb{R}^N $

Manli Song^1, and
Jinggang Tan^2,3, ,

1.
School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710129, China
2.
Departamento de Matemática, Universidad Técnica Federico Santa María, Avda. España 1680, Valparaíso, Chile
3.
Department of Mathematics, Jianghan University, Wuhan, Hubei, 430056, China

^* Corresponding author
^* Corresponding author

Received: July 31, 2019

Revised: February 29, 2020

Published: June 2020

M. Song is supported by the China Postdoctoral Science Foundation (Grant No. 2017M623230), the National Natural Science Foundation of China (Grant No. 11701452), the Natural Science Foundation of Shaanxi Province (Grant No. 2020JQ-112) and the Fundamental Research Funds for the Central Universities (Grant No. 3102018zy041). J. Tan is partially supported by Chile Government grant FONDECYT grant 1160519 and by MINECO grants MTM2014-52402-C3-1-P and MTM2017-84214-C2-1-P (Spain Government Grants)

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Abstract

We establish uncertainty principles and Hardy inequalities for the fractional Grushin operator, which are reduced to those inequalities for the fractional generalized sublaplacian. The key ingredients to obtain them are an explicit integral representation and a ground state representation for the fractional powers of generalized sublaplacian.

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Mathematics Subject Classification: Primary: 35H20, 26D10.

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