"Small step" remodeling and counterexamples for weighted estimates with arbitrarily "smooth" weights

For an $A_p$ weight $w$ the norm of the Hilbert Transform in $L^p(w)$, $1<p<\infty$ is estimated by $[w]_{A_p}^{s}$, where $[w]_{A_p}$ is the $A_p$ characteristic of the weight $w$ and $s = \max(1,1/(p-1))$; as simple examples with power weights show, these estimates are sharp. A natural question to ask, is whether it is possible to improve the exponent $s$ in the above estimate if one replaces the $A_p$ characteristic by its "fattened" version, where the averages are replaced by Poisson-like averages. For power weights (for example with $p=2$ and Poisson averages) one can see that there is indeed an improvement in the exponent: but is it true for general weights? In this paper we show that the optimal exponent $s$ remains the same by constructing counterexamples for arbitrarily "smooth" weights (in the sense that the doubling constant is arbitrarily close to $2$), so the "fattened" $A_p$ characteristic is equivalent to the classical one, and such that $\|T\|_{L^p(w)} \sim [w]_{A_p}^{s}$. We use the ideas from the unpublished manuscript by F. Nazarov disproving Sarason's conjecture. We start from simple classical counterexamples for dyadic models, and then by using what we call "small step construction" we transform them into examples with weights that are arbitrarily dyadically smooth. F.~Nazarov had used Bellman function method to prove the existence of such examples, but our construction gives a way to get such examples from the standard dyadic ones. We then use a modification of "remodeling", introduced by J.~Bourgain and developed by F.~Nazarov, to get from examples for dyadic models to examples for the Hilbert transform. As an added bonus, we present a proof that the $L^p$ analog of Sarason's conjecture is false for all $p$, $1<p<\infty$.