Given a doubling measure \(\mu\) on \(\mathbb R^n\), it is well known that Calderón-Zygmund operators which are bounded on \(L^2(\mu)\) are also of weak type \((1,1)\). Recently, it has been shown that the same result holds if the doubling condition on \(\mu\) is replaced by a mild growth condition on \(\mu\): \[ \mu(B(x, r))\leq Cr^m \text{ for all }x\in \mathbb R^n \text{ and }r> 0, \] where \(m\) is some fixed number with \(0< m\leq n\). More details can be found in [F. Nazarov, S. Treil and A. Volberg, Int. Math. Res. Not. 1998, No. 9, 463–487 (1998; Zbl 0918.42009)].
In the paper under review, the author gives an alternative proof of the above result. His 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.