MAJORIZATION OF SINGULAR INTEGRAL OPERATORS WITH CAUCHY KERNEL ON L2

Let a, b, c and d be functions in L-2 = L-2(T, d theta/2 pi), 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-Szego type set (HS)(r) to establish the condition of a, b, c and d satisfying parallel to Af parallel to(2) >= parallel to Bf parallel to(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.