A Weak-Type Estimate for Commutators

Let K be a smooth Calderón–Zygmund convolution kernel on R² and suppose that we are given a function $a\in L^\infty$. The two-dimensional commutator $$Tf(x)=\int K(x-y)\ f(y)\int_{[x, y]}\!\!\!\!\!\!\!\!\!\!\!-\ \ \ a(z){\mathrm dzd}y$$ was shown to be bounded on L^p(R²), p>1 by Christ and Journé [2]. In this article, we show that this operator is also of weak type (1,1).