### $BMO$ estimates for nonvariational operators with discontinuous coefficients structured on Hörmander's vector fields on Carnot groups

#### Abstract

We consider the class of operators $Lu=\sum_{i,j=1}^{q}a_{ij}(x)X_{i}X_{j}u,$ where $X_{1},X_{2},\dots,X_{q}$ are homogeneous left-invariant Hörmander's vector fields on $\mathbb{R}^{N}$ with respect to a structure of Carnot group, $q\leq N,$ the matrix $\{ a_{ij}\}$ is symmetric and uniformly positive on $\mathbb{R}^{q},$ the coefficients $a_{ij}$ belong to $L^{\infty}\cap VLMO_{loc}( \Omega)$ ("vanishing logarithmic mean oscillation") with respect to the distance induced by the vector fields (in particular, they can be discontinuous), and $\Omega$ is a bounded domain of $\mathbb{R}^{N}$. We prove local estimates in $BMO_{loc}\cap L^{p}$ of the following kind: \begin{align*} & \Vert X_{i}X_{j}u\Vert _{BMO_{loc}^{p}( \Omega^{\prime }) }+\Vert X_{i}u\Vert _{BMO_{loc}^{p}( \Omega ^{\prime}) } \\ & \leq c\big\{ \Vert Lu\Vert _{BMO_{loc}^{p}( \Omega) }+\Vert u\Vert _{BMO_{loc}^{p}( \Omega) }\big\} \end{align*} for any $\Omega^{\prime}\Subset\Omega$, $1 < p < \infty$. Even in the uniformly elliptic case $X_{i}=\partial_{x_{i}}$, $q=N$ our estimates improve the known results.

#### Article information

Source
Adv. Differential Equations Volume 18, Number 9/10 (2013), 955-1004.

Dates
First available in Project Euclid: 2 July 2013

Bramanti, Marco; Fanciullo, Maria Stella. $BMO$ estimates for nonvariational operators with discontinuous coefficients structured on Hörmander's vector fields on Carnot groups. Adv. Differential Equations 18 (2013), no. 9/10, 955--1004. https://projecteuclid.org/euclid.ade/1372777765