Divergent operator with degeneracy and related sharp inequalities

TL;DR

This paper classifies extremal functions for a sharp weighted Sobolev inequality involving degenerate divergent operators, deriving related inequalities and classifying extremals for specific parameters.

Contribution

It provides a complete classification of extremal functions for a new class of sharp inequalities involving degenerate operators on the upper half space.

Findings

Classified all positive extremal functions for the weighted Sobolev inequality.

Derived a sharp Sobolev inequality involving Baouendi-Grushin operator.

Classified extremal functions for all >0 with ma0e02 or na0e11.

Abstract

In this paper we classify all positive extremal functions to a sharp weighted Sobolev inequality on the upper half space, which involves divergent operators with degeneracy on the boundary. As an application of the results, we can derive a sharp Sobolev type inequality involving Baouendi-Grushin operator, and classify certain extremal functions for all $\tau>0$ and $m\ne2 $ or $ n\ne1$.