Speaker: Zhenzhen Wei, Memorial University

 

Title: Caffarelli-Kohn-Nirenberg inequalities on homogeneous groups and for Minkowski functionals

 

Abstract: Caffarelli-Kohn-Nirenberg (CKN) inequalities form a fundamental family of weighted interpolation inequalities and are closely related to several classical inequalities, including the Hardy inequality and the Heisenberg uncertainty principle. In this talk, I will discuss sharp CKN inequalities in two geometric settings: on homogeneous groups and in Euclidean spaces with anisotropy induced by Minkowski functionals.

 

I will first present sharp $L^2$-CKN inequalities on homogeneous groups, including the sharp constants and extremal functions. I will then turn to Minkowski functionals generated by smooth, strictly convex bodies, without assuming origin symmetry. In this setting, I will describe exact remainder identities that lead to sharp anisotropic $L^2$ and $L^p$ inequalities, together with the corresponding extremal functions and critical non-attainment results. Applications to anisotropic Heisenberg uncertainty principles and gradient inequalities will also be discussed. Finally, I will present quantitative stability estimates for the anisotropic CKN inequalities, obtained by reducing the remainder terms to one-dimensional weighted Poincar\'e inequalities.

 

The main theme is how sharp Euclidean inequalities extend when the Euclidean norm is replaced by a homogeneous quasi-norm or by a Minkowski functional, while retaining explicit sharp constants, extremals, and stability properties.

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Location: Online

Date and Time: Friday, Oct. 2 at 11:00 AM - 12:00 PM (NDT)

For more information: https://mun.webex.com/mun/j.php?MTID=me93c25daa67127cc995ce18e9d1ec601

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