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 $\mu(B(x,r)) \le Cr^n$, $n \in (0, d]$. 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 $p, q \in [2-\epsilon(n), 2]$, $\epsilon(n) > 0$. Consequences for (non-homogeneous) local $Tb$ theorems are discussed.