A proof of the weak (1,1) inequality for singular integrals with non doubling measures based on a Calderón-Zygmund decomposition

Given a doubling measure $\mu$ on ${\mathbb R}^d$, it is a classical result of harmonic analysis that Calderón-Zygmund operators which are bounded in $L^2(\mu)$ are also of weak type (1,1). Recently it has been shown that the same result holds if one substitutes the doubling condition on $\mu$ by a mild growth condition on $\mu$. In this paper another proof of this result is given. The proof is very close in spirit to the classical argument for doubling measures and it is based on a new Calderón-Zygmund decomposition adapted to the non doubling situation.