Improved Cotlar's inequality in the context of local Tb theorems

In the context of local Tb theorems with $L^p$ testing conditions we prove an enhanced Cotlar's inequality. This is related to the problem of removing the so called buffer assumption of Hytönen–Nazarov, which is the final barrier for the full solution of S. Hofmann's problem. We also investigate the problem of extending the Hytönen–Nazarov result to non-homogeneous measures. We work not just with the Lebesgue measure but with measures μ in ${\mathbb{R}}^d$ satisfying $\mu (B(x,r))\le C{r}^n$, $n\in (0,d]$. The range of exponents in the Cotlar type inequality depend on n. Without assuming buffer we get the full range of exponents $p,q\in (1,2]$ for measures with $n\le 1$, and in general we get $p,q\in 2??(n),2]$, $?(n)>0$. Consequences for (non-homogeneous) local Tb theorems are discussed.