A Sobolev-like inequality for the Dirac operator

TL;DR

This paper establishes a Sobolev-like inequality for the Dirac operator on compact spin manifolds, providing new tools for analyzing nonlinear equations involving the Dirac operator with critical Sobolev exponents.

Contribution

It introduces a nearly optimal Sobolev inequality for the Dirac operator and applies it to determine solution existence for related nonlinear equations.

Findings

Proved a Sobolev-like inequality with near-optimal constant.

Derived a criterion for solutions to nonlinear Dirac equations.

Identified a specific case where the nonlinear equation can be solved.

Abstract

In this article, we prove a Sobolev-like inequality for the Dirac operator on closed compact Riemannian spin manifolds with a nearly optimal Sobolev constant. As an application, we give a criterion for the existence of solutions to a nonlinear equation with critical Sobolev exponent involving the Dirac operator. We finally specify a case where this equation can be solved.