Weak type $(1,1)$ estimates for Riesz transforms of elliptic operators and magnetic Schrödinger operators

Abstract

Let $M$ be a complete Riemannian manifold and define a magnetic Schrödinger operator $\LL = ({\bf d}+ iA)^*({\bf d}+iA) + V$ where $A$ is a real $1$-differential form in $L_{loc}^2$ and $V\in L_{loc}^1$ is a nonnegative electric potential. We prove that the magnetic Riesz transform $({\bf d} + iA) \LL^{-\einhalb}$ is of weak type $(1,1)$ with bound at most $2$. It extends to a bounded operator on $L^p$ for $p\in (1,2]$ with norm bounded by a constant $c_p$ depending only on $p$. It is then independent of the dimension of the manifold and independent of $A$ and $V$. We also prove weak type $(1,1)$ bounds for symmetric elliptic operators subject to various boundary conditions on rough domains of $\RR^d$. The proofs are based on a new weak type $(1,1)$ criterion for operators.