Majorization of Singular Integral Operators with Cauchy Kernel on L

Let a, b, c and d be functions in L = L(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, i.e., A∗A ≥ B∗B on L. Applications are then presented for the majorization of singular integral operators on weighted L spaces, and for the normal singular integral operators aP + bQ on L when a − b is a complex constant. Department of Mathematics, Hokkai-Gakuen University, Sapporo 062-8605, Japan. E-mail address: [email protected] Date: Received: 1 July 2013; Accepted: 16 September 2013. 2010 Mathematics Subject Classification. Primary 45E10; Secondary 47B35.