Weak type $(1,1)$ bounds for Riesz transforms for elliptic operators in non-divergence form

TL;DR

This paper proves weak type (1,1) bounds for Riesz transforms associated with elliptic operators in non-divergence form under certain conditions on the weight, extending their boundedness in weighted L^p spaces.

Contribution

It establishes weak (1,1) bounds for Riesz transforms of elliptic operators with smooth coefficients in non-divergence form, under the assumption that the weight belongs to A_2.

Findings

Riesz transforms are of weak type (1,1) with respect to measure W(x)dx.

Boundedness of Riesz transforms in weighted L^p spaces for 1<p<2.

Applicability to operators with coefficients having small BMO norm.

Abstract

Let $L=-\sum_{i,j=1}^n a_{ij}D_iD_j$ be the elliptic operator in non-divergence form with smooth real coefficients satisfying uniformly elliptic condition. Let $W$ be the global nonnegative adjoint solution. If $W\in A_2$, we prove that the Riesz transforms $\nabla L^{-\frac{1}{2}}$ is of weak type $(1,1)$ with respect to the measure $W(x)dx$. This, together with $L^2_W$ boundedness of Riesz transforms \cite{EHH}, implies that the Riesz transforms are bounded in $L^p_W$ for $1<p<2$. Our results are applicable to the case of real coefficients having sufficiently small BMO norm.