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

We prove in the context of local $Tb$ theorems with $L^p$ type testing conditions an improved version of Cotlar’s inequality. This is related to the problem of removing the so called buffer assumption of Hyt"onen-Nazarov, which is the final barrier for the full solution of S. Hofmann’s problem. We also investigate the problem of extending the Hyt"onen-Nazarov result to non-homogeneous measures. We work not just with the Lebesgue measure but with measures $\mu$ in ${\mathbb{R}}^d$ satisfying $\mu (B(x,r))\le C{r}^n$, $n \in (0, d]$\mathrm{.}TherangeofexponentsintheCotlartypeinequalitydependon$n$. 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 $p,q\in [2-ϵ(n),2]$, $ϵ(n)>0$. Consequences for (non-homogeneous) local $Tb$ theorems are discussed.