Title: Quadratic $A^1$ bounds for commutators of singular integrals with BMO functions
Authors: Carmen Ortiz-Caraballo
Issue: Volume 60 (2011), Issue 6, 2107-2130
Abstract:
![For any Calder\'on-Zygmund operator $T$ and any $\mathrm{BMO}$ function $b$ we prove the following quadratic estimate:
\[
\big\|[b,T]\big\|_{L^p(w)} \le c\|b\|_{\mathrm{BMO}}(pp')^2\big[w\big]_{A_1}^2, 1 < p < \infty, w \in A_1,
]
with constant $c = c(n,T)$ being the estimate optimal on $p$ and the exponent of the weight constant. As an endpoint estimate we prove
\begin{align*}
{}& w\big(\big\{x\in\mathbb{R}^n : \big|[b,T]f(x)\big| > \lambda\big\})\\
{}& \qquad \leq c\Phi([w]_{A_1})^2\int_{\mathbb{R}^n}\Phi\left(\frac{|f(x)|}{\lambda}\right)w(x) \mathrm{d}x,
\end{align*}
where $\Phi(t) = t(1+\log^{+}t)$ and constant $c = c(n,T,\|b\|_{\mathrm{BMO}})$.](https://iumj.s3-us-west-2.amazonaws.com/abstracts/4494.png)
Title: Quadratic $A^1$ bounds for commutators of singular integrals with BMO functions
Authors: Carmen Ortiz-Caraballo
Issue: Volume 60 (2011), Issue 6, 2107-2130
Abstract: