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

Abstract

Let $L=-{\sum }_{i,j=1}^n{a}_{ij}{D}_i{D}_j$ 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 $L_W^2$ boundedness of Riesz transforms \cite{EHH}, implies that the Riesz transforms are bounded in $L_W^p$ for $1<p<2$. Our results are applicable to the case of real coefficients having sufficiently small BMO norm.

15 pages