Endpoint estimates for commutators of singular integrals on spaces of homogeneous type (Q1399856)

Let \((X,d,\mu)\) be a space of homogeneous type (\(d\) is a quasimetric on \(X\) and \(\mu\) is a non negative Borel measure on \(X\)). The BMO space of \(X\) is \(BMO(X,\mu)=\{f\in L_{loc}^{1}(X): \|f\|_{*}<+\infty\}\) where \(\|f\|_{*}=Sup_{B}\frac{1}{\mu(B)}\int_{B}\mid f(y)-f_{B} \mid d\mu(y)\) and \(f_{B}=\frac{1}{\mu(B)}\int_{B}f(x)d\mu(x)\) for a ball \(B\) in \(X\). The main result of the paper under review is stated as follows: let \(T\) be a singular integral operator which is bounded on \(L^{2}(X,\mu)\) and \(b\in BMO(X,\mu)\). Then there exists a positive constant \(C\) such that for each bounded function \(f\) with bounded support and for all \(\lambda >0\) \[ \mu\big(\{y\in X: \mid[b,T]f(y)\mid>\lambda\}\big)\leq C\|b\|_{*}\int_{X}\frac{\mid f(y)\mid}{y}(1+log^{+}(\frac{\mid f(y)\mid}{y})) d\mu(y) \] where \([b,T]\) is the commutator of \(b\) and \(T\).