The sharp maximal function approach to $L^{p}$ estimates for operators structured on H\"{o}rmander's vector fields

TL;DR

This paper introduces a new proof technique for $L^{p}$ estimates of second derivatives of degenerate elliptic operators structured on Hörmander's vector fields, extending Krylov's method to a Carnot group setting.

Contribution

It provides a novel proof of interior $L^{p}$ estimates using sharp maximal functions, applicable to operators on Carnot groups with VMO coefficients, extending previous results.

Findings

Established interior $L^{p}$ estimates for second derivatives

Extended Krylov's technique to Carnot group context

Provided a new proof approach for degenerate elliptic operators

Abstract

We consider a nonvariational degenerate elliptic operator structured on a system of left invariant, 1-homogeneous, H\"ormander's vector fields on a Carnot group in $R^{n}$, where the matrix of coefficients is symmetric, uniformly positive on a bounded domain of $R^{n}$ and the coefficients are bounded, measurable and locally VMO in the domain. We give a new proof of the interior $L^{p}$ estimates on the second order derivatives with respect to the vector fields, first proved by Bramanti-Brandolini in [Rend. Sem. Mat. dell'Univ. e del Politec. di Torino, Vol. 58, 4 (2000), 389-433], extending to this context Krylov' technique, introduced in [Comm. in P.D.E.s, 32 (2007), 453-475], consisting in estimating the sharp maximal function of the second order derivatives.