On the necessity of the constant rank condition for $L^p$ estimates

TL;DR

This paper proves that the constant rank condition is necessary for certain elliptic $L^p$ estimates of linear operators with non-trivial kernels, extending classical results.

Contribution

It establishes the necessity of the constant rank condition for $L^p$ estimates, generalizing previous sufficiency results.

Findings

Constant rank condition is necessary for $L^p$ estimates.

Generalization of classical elliptic estimate results.

Extension to operators with non-trivial kernels.

Abstract

We consider a generalization of the elliptic $L^p$-estimate suited for linear operators with non-trivial kernels. A classical result of Schulenberger and Wilcox (Ann. Mat. Pura Appl. (4) 88: 229-305, 1971) shows that if the operator has constant rank then the estimate holds. We prove necessity of the constant rank condition for such an estimate.