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

Let $M$ be a complete Riemannian manifold and define a magnetic Schrödinger operator $\text{LL}=(\mathbf{d}+iA{)}^{\ast }(\mathbf{d}+iA)+V$ 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 $(\mathbf{d}+iA){\text{LL}}^{-\text{einhalb}}$ 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 ${\text{RR}}^d$. The proofs are based on a new weak type $(1,1)$ criterion for operators.