On horizontal Hardy, Rellich, Caffarelli-Kohn-Nirenberg and $p$-sub-Laplacian inequalities on stratified groups

Abstract

In this paper, we present a version of horizontal weighted Hardy-Rellich type and Caffarelli-Kohn-Nirenberg type inequalities on stratified groups and study some of their consequences. Our results reflect on many results previously known in special cases. Moreover, a new simple proof of the Badiale-Tarantello conjecture [2] on the best constant of a Hardy type inequality is provided. We also show a family of Poincaré inequalities as well as inequalities involving the weighted and unweighted $p$-sub-Laplacians.

20 pages

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• Caffarelli-Kohn-Nirenberg and Sobolev type inequalities on stratified Lie groups
• Global existence and blow-up of solutions to porous medium equation and pseudo-parabolic equation, I. Stratified Groups
• $L^p$-Caffarelli-Kohn-Nirenberg type inequalities on homogeneous groups
• On Green functions for Dirichlet sub-Laplacians on H-type groups
• Uncertainty relations on nilpotent Lie groups
• A comparison principle for nonlinear heat Rockland operators on graded groups
• Some functional inequalities for the fractional p-sub-Laplacian
• Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability, and superweights for $L^p$-weighted Hardy inequalities
• Hardy and Rellich inequalities for anisotropic p-sub-Laplacians
• A simple proof of the Hardy inequality on Carnot groups and for some hypoelliptic families of vector fields
• Hardy-Littlewood, Bessel-Riesz, and fractional integral operators in anisotropic Morrey and Campanato spaces
• Representation formulae for the higher-order Steklov and $L^{2^m}$-Friedrichs inequalities
• Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group