An Example of a Singular Integral and a Weight

Abstract

Abstract Sharp weighted inequalities were recently proved for several classical operators in Harmonic analysis, however for the rough singular integral the sharp result remains open. The best bound so far was found by Hytönen, Roncal and Tapiola in [2]. For $A_2$ weight, it is quadratic, meaning $\|T_\Omega f\|_{L^2_{\omega }}\leq C [\omega ]_2^2 \|f\|_{L^2_{\omega }}.$ The authors also conjectured that the best bound is linear. We provide example of $A_2$ weights $\omega _n$, test functions $f_n$ and rough singular integrals $$ \begin{align*} & T_{\Omega_n}(f)(x)=\textrm{p.v.} \int_{\mathbb R^2} |y|^{-2} \Omega_n(y/|y|)f(x-y) dy, \end{align*}$$where $\Omega _n $ is a function in $L^\infty (\mathbb S^{1})$ with norm $1$ and vanishing integral such that $$ \begin{align*} &\|T_{\Omega_n} f_n\|_{L^2_{\omega_n}}\geq C [\omega_n]_2^{3/2} \|f_n\|_{L^2_{\omega_n}}\end{align*}$$and $[\omega _n]_2\approx n,$ disproving the conjecture.