Paper detail

"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 &#34;fattened&#34; 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 &#34;smooth&#34; weights (in the sense that the doubling constant is arbitrarily close to $2$), so the &#34;fattened&#34; $A_p$ 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 We use the ideas from the unpublished manuscript by F. Nazarov disproving Sarason&#39;s conjecture. We start from simple classical counterexamples for dyadic models, and then by using what we call &#34;small step construction&#34; 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 &#34;remodeling&#34;, 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&#39;s conjecture is false for all $p$, $1<p<\infty$.