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

TL;DR

This work establishes weak type $(1,1)$ bounds for the Riesz transform $\nabla L^{-1/2}$ associated to a second-order elliptic operator in non-divergence form under a Muckenhoupt $A_2$ condition on the global adjoint solution $W$. Building on the Kato square root results and the weighted $L^2$-calculus, the authors prove $L^p_W$-boundedness of the Riesz transform for $1<p<2$ by combining a Calderón–Zygmund decomposition with precise gradient heat-kernel estimates. A central technical achievement is a sharp gradient bound for the heat kernel, which feeds into off-diagonal decay and the treatment of the non-local part of the operator. The paper also develops Hardy space theory for the operator $L$, obtaining boundedness of $\nabla L^{-1/2}$ on the associated Hardy spaces and interpolating to $L^p_W$-spaces for $1<p<2$, with applications to coefficients of small BMO. These results extend endpoint and weighted estimates for Riesz transforms in the non-divergence setting and relate to the Kato-square-root program in weighted contexts.

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.

Theorems & Definitions (15)

• Theorem 1.1: EHH
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Proposition 3.1
• proof