Majorization of singular integral operators with Cauchy kernel on \(L^{2}\) (Q376641)

The author considers Cauchy singular integral operators on the unit circle \(S^1\), defined as standard in terms of the projections \(P= (I+ C)/2\), \(Q= (I- C)/2\), where \(C\) is the classical Cauchy operator on \(S^1\). Namely, let \(A= aP+ bQ\), \(B= cP+ dQ\), where \(a,b,c,d\in L^2(S^1)\), hence \(A\) and \(B\) are not in general bounded on \(L^2(S^1)\). Precise necessary and sufficient conditions are given for the validity of the estimates \[ \| Af\|_2\geq \| Bf\|_2, \] for an arbitrary trigonometric polynomial \(f\). The conditions involve the Helson-Szegö class of functions \(HS\), see [\textit{J. B. Garnett}, Bounded analytic functions. Pure and Applied Mathematics, 96. New York etc.: Academic Press, A subsidiary of Harcourt Brace Javanovich, Publishers. (1981; Zbl 0469.30024)].