Majorization of Singular Integral Operators with Cauchy Kernel on $L^2$

Abstract

Let a, b, c and d be functions in L 2 = L 2 (T, dθ/2π), where T denotes the unit circle. Let P denote the set of all trigonometric polynomials. Suppose the singular integral operators A and B are defined by A = aP + bQ and B = cP + dQ on P, where P is an analytic projection and Q = I − P is a co-analytic projection. In this paper, we use the Helson-Szegő type set (HS)(r) to establish the condition of a, b, c and d satisfying Af 2 ≥ Bf 2 for all f in P. If a, b, c and d are bounded measurable functions, then A and B are bounded operators, and this is equivalent to that B is majorized by A on L 2 , i.e., A * A ≥ B * B on L 2 . Applications are then presented for the majorization of singular integral operators on weighted L 2 spaces, and for the normal singular integral operators aP + bQ on L 2 when a − b is a complex constant.