Continuity of weighted estimates in \(A_p\) norm (Q2845553)

For \(1<p<\infty\), let \(\omega>0\) be an \(A_p\)-weight on \(\mathbb R^n\) and let \(L_p(\omega)\) be the associated weighted \(L^p\) space. Note that the \(A_p\)-norm is given by NEWLINE\[NEWLINE||\omega||_{A_p}=\sup_Q \left( \frac{1}{|Q|}\int_Q \omega dx\right) \left( \frac{1}{|Q|}\int_Q \omega^{-(p-1)^{-1}}dx\right)^{p-1}, NEWLINE\]NEWLINE where \(Q\) runs through the set of cubes in \(\mathbb R^n\). The authors show that, if \(T\) is a linear operator acting on the usual \(L^p(\mathbb R^n)\) space for which there exists an increasing positive function \(F\) such that, for any \(A_p\)-weight, NEWLINE\[NEWLINE||T||_{L_p(\omega)\to L_p(\omega)}\leq c F(||\omega||_{A_p}),NEWLINE\]NEWLINE then NEWLINE\[NEWLINE\lim_{||\omega||_{_{A_p}}\to 1} ||T||_{L_p(\omega)\to L_p(\omega)}= ||T||_{L^p(\mathbb R^n)\to L^p(\mathbb R^n)}.NEWLINE\]NEWLINE Precise estimations of \(||T||_{L_p(\omega)\to L_p(\omega)}\) are given, too. The cases where \(T\) is the Hilbert transform or the Riesz projection are investigated in detail.