Caffarelli-Kohn-Nirenberg inequalities for curl-free vector fields and second order derivatives

TL;DR

This paper extends Caffarelli-Kohn-Nirenberg inequalities to curl-free vector fields with radial weights and introduces new sharp second order inequalities for scalar fields, providing explicit minimizers and broadening the theoretical framework.

Contribution

It generalizes CKN inequalities to curl-free vector fields and introduces new second order inequalities for scalar fields with explicit minimizers.

Findings

Extended CKN inequalities to curl-free vector fields with sharp constants.

Derived new second order inequalities for scalar fields with radial weights.

Identified explicit minimizers using solutions of Kummer's differential equations.

Abstract

The present work has as a first goal to extend the previous results in \cite{CFL20} to weighted uncertainty principles with nontrivial radially symmetric weights applied to curl-free vector fields. Part of these new inequalities generalize the family of Caffarelli-Kohn-Nirenberg (CKN) inequalities studied by Catrina and Costa in \cite{CC} from scalar fields to curl-free vector fields. We will apply a new representation of curl-free vector fields developed by Hamamoto in \cite{HT21}. The newly obtained results are also sharp and minimizers are completely described. Secondly, we prove new sharp second order interpolation functional inequalities for scalar fields with radial weights generalizing the previous results in \cite{CFL20}. We apply new factorization methods being inspired by our recent work \cite{CFL21}. The main novelty in this case is that we are able to find a new independent family of minimizers based on the solutions of Kummer's differential equations. We point out that the two types of weighted inequalities under consideration (first order inequalities for curl-free vector fields vs. second order inequalities for scalar fields) represent independent families of inequalities unless the weights are trivial.