Weighted norm inequalities for some singular integral operators (Q1368846)

Let \[ Sf(\zeta)= {1\over\pi i} \int_T {f(z)\over z-\zeta} dz \] be the singular integral operator on the unit circle \(T\). For functions \(\alpha\) and \(\beta\) in \(L^\infty\) set \[ S_{\alpha,\beta}= {\alpha- \beta\over 2} S+ {\alpha+ \beta\over 2} I, \] where \(I\) is the identity operator. For a nonnegative function \(W\) in \(L^1\) let \[ \| f\|_W= \Biggl\{ \int_T| f|^2 W du\Biggr\}^{1/2}. \] The authors obtain conditions in order that the estimates \[ \| S_{\alpha, \beta}f\|_W\leq C\| f\|_W,\quad \| S_{\alpha, \beta}f\|_W\geq \delta\| f\|_W, \] hold for \(f\in A+\overline A_0\), where \(A\) is the disc algebra, that is, the algebra of all continuous functions in \(T\) whose negative Fourier coefficients vanish, and \(\overline A_0\) is the set of all complex conjugates of functions in \(A_0\), the subspace of all functions in \(A\) whose mean value is zero. For that purpose they introduce a class of weights generalizing that by Helson and Szegö. The authors also obtain a necessary and sufficient condition for the existence of a weight \(U\) such that the inequality \[ \| S_{\alpha, \beta}f\|_U\leq C\| f\|_W,\quad\text{or}\quad \| S_{\alpha, \beta}f\|_W\geq \delta\| f\|_U, \] holds for all \(f\in A+\overline A_0\).