An extrapolation theorem with applications to weighted estimates for singular integrals (Q425728)

Let \(T\) be a Calderón-Zygmund operator. The weak Muckenhoupt-Wheeden conjecture says that NEWLINE\[NEWLINE\|Tf\|_{L^{1,\infty}(w)} \leq c(T,n)\|w\|_{A_1}\|f\|_{L^1(w)}. NEWLINE\]NEWLINE This is still open even for the Hilbert transform. \textit{A. K. Lerner, S. Ombrosi} and \textit{C. Pérez} [Math. Res. Lett. 16, No. 1, 149--156 (2009; Zbl 1169.42006)] showed that, for any Calderón-Zygmund operator, NEWLINE\[NEWLINE\|Tf\|_{L^{1,\infty}(w)} \leq c(T,n)\|w\|_{A_1} \log(1+\|w\|_{A_1}) \|f\|_{L^1(w)}. NEWLINE\]NEWLINE In this paper, the authors prove an extrapolation theorem saying that the weighted weak type \((1, 1)\) inequality for \(A_1\) weights implies the strong \(L^p(w)\) bound in terms of the \(L^p(w)\) operator norm of the maximal operator \(M\). This along with the weak Muckenhoupt-Wheeden conjecture leads the authors to conjecture that the estimate NEWLINE\[NEWLINE\|T\|_{L^p(w)}\leq c(T,n,p)\|M\|^p_{L^p(w)} NEWLINE\]NEWLINE holds for any Calderón-Zygmund operator \(T\) and any \(1<p<\infty\). The latter conjecture yields sharp estimates for \(\|T\|_{L^p(w)}\) in terms of the \(A_q\) characteristic of \(w\) for any \(1<q <p\). The authors get a weaker inequality NEWLINE\[NEWLINE\|T\|_{L^p(w)}\leq c\|M\|^p_{L^p(w)}\log(1+\|M\|_{L^p(w)}) NEWLINE\]NEWLINE with the corresponding estimates for \(\|w\|_{A_q}\) when \(1<q <p\).