Quadratic $A^1$ bounds for commutators of singular integrals with BMO functions Carmen Ortiz-Caraballo 42B2042B2546B7047B38commutatorssingular integralsBMOA1Ap 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 $c = c(n,T,\|b\|_{\mathrm{BMO}})$. Indiana University Mathematics Journal 2011 text pdf 10.1512/iumj.2011.60.4494 10.1512/iumj.2011.60.4494 en Indiana Univ. Math. J. 60 (2011) 2107 - 2130 state-of-the-art mathematics http://iumj.org/access/