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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 exampleswith 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 seethat there is indeed an improvement in the exponent: but is it true for generalweights?  In this paper we show that the optimal exponent $s$ remains the same byconstructing counterexamples for arbitrarily "smooth" weights (in the sensethat 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  We use the ideas from the unpublished manuscript by F. Nazarov disprovingSarason's conjecture. We start from simple classical counterexamples for dyadicmodels, and then by using what we call "small step construction" we transformthem into examples with weights that are arbitrarily dyadically smooth.F.~Nazarov had used Bellman function method to prove the existence of suchexamples, but our construction gives a way to get such examples from thestandard dyadic ones. We then use a modification of "remodeling", introduced byJ.~Bourgain and developed by F.~Nazarov, to get from examples for dyadic modelsto examples for the Hilbert transform.  As an added bonus, we present a proof that the $L^p$ analog of Sarason'sconjecture is false for all $p$, $1<p<\infty$.

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