Sharp weighted weak-type bounds for commutators of singular integrals

We establish sharp weak-type bounds for commutators of Calderón–Zygmund operators and $\text{BMO}({\mathbb{R}}^n)$ symbols with respect to $A_p$ weights. We prove that the bound \looseness=-1 $$\mathrm{\parallel }[b,T]{\mathrm{\parallel }}_{L^p(w)\to L^{p,\mathrm{\infty }}(w)}\lesssim [w{]}_{A_p}^{\max (2,p^{\mathrm{\prime }})}$$ for $p\in (1,\mathrm{\infty })$ and $w\in A_p$ is sharp for general $T\in \text{CZO}({\mathbb{R}}^n)$ and $b\in \text{BMO}({\mathbb{R}}^n)$. In the case of the standard logarithmic symbol, we show that the estimate improves to $$\mathrm{\parallel }[\log \mathrm{\mid }\cdot \mathrm{\mid },T]{\mathrm{\parallel }}_{L^p(w)\to L^{p,\mathrm{\infty }}(w)}\lesssim [w{]}_{A_p}^{\max (2,\frac{1}{p}+\frac{p^{\mathrm{\prime }}}{p})}$$ and that this dependence is optimal for this symbol. Our upper bound for the logarithmic symbol follows from the pointwise control of the commutator by a sum of the Hardy–Littlewood maximal operator and the squares of the radial Hardy operator and its adjoint, while our lower bounds are consequences of a general result for singular integral operators.