THE STUDY OF SINGULAR INTEGRAL OPERATORS RELATED TO THE PREDICATION THEORY AND THE UNIFORM ALGEBRA

设和是单位圆T上的有界可测函数，设W是T上的正权函数，设P_+是加权空间L^2 （W，T）上的Riesz投影。一维线性奇异积分算子S_< α, β >定义为S_< α, β >f= α P_+f+ β ap_ -f, （f*L^2(W，T)）其中P_-=I-P_+。该算子与Toeplitz算子和Hankel算子相关。首先，我们研究了S_<, >在L^2 （W，T）上的有界性。由Koosis定理可知，有很多，W使得P_+无界而S_<, >有界。接下来，我们研究了S_<, >的范数。设h为外函数W=|h|^2，设为单模函数使=h^^ /h。通过希尔伯特空间论证和柯拉-萨多斯基提升定理，我们得到了三个公式来计算S_<, >在L^2 （W，T）上使用，和的范数。第一个公式给出了实变量的凸函数，实变量的不动点是S_<, >在L^2 （W，T）上的范数。第二个公式类似于Feldman-Krupnik-Marcus公式（参见gohberg - krupnik的书）。第三个公式和他们的公式很不一样。如果W是一个常数，那么就变成了一个常数，因此我们的结果变得更简单，如下所示。让m =(α| | ^ 2 +β| | ^ 2)/ 2,n = | |α| ^ 2 -β| | | / 2 ^ 2。然后m + n = max{α| | ^ 2,β| | ^ 2}和m n =分钟{α| | ^ 2,β| | ^ 2}。设||f||表示f的L^2(T)范数，设H^*(T)表示Hardy空间。那么下列等式成立。sup{||S_<alpha, beta>f||^2/||f||^2；f * L ^ 2 (T)} = inf {ess。sup_T (m + (| alphabeta ^ ^ _-k | ^ 2 + n ^ 2) ^ < 1/2 >);k * H ^ * (T)}。{||S_<alpha, beta>f||^2/||f||^2；f * L ^ 2 (T)} = {ess一同坐席。inf_T (m - (| alphabeta ^ ^ _-k | ^ 2 + n ^ 2) ^ < 1/2 >);k * H ^ * (T)}。

Let alpha and beta be bounded measurable functions on the unit circle T, let W be a positive weight function on T, and let P_+ be the Riesz projection on the weighted space L^2 (W,T). The one-demensional linear singular integral operator S_<alpha, beta> is defined by S_<alpha, beta>f=alphaP_+f+betaP_-f, (f*L^2(W,T)) where P_-=I-P_+. This operator is related to Toeplitz operators and Hankel operators. At first, we study the boundedness of S_<alpha, beta> on L^2 (W,T). It follows from the Koosis theorem that there are many alpha, beta, W such that P_+ is not bounded and S_<alpha, beta> is bounded. Next, we study the norm of S_<alpha, beta>. Let h be an outer function such that W=|h|^2, and let phi be an unimodular function such thatphi=h^^_/h. By the Hilbert space argument and the Cotlar-Sadosky's lifting theorem, we get three formulas to compute the norm of S_<alpha, beta> on L^2 (W,T) using alpha, beta and phi. The first formula gives the convex function of a real variable whose fixed point is the norm of S_<alpha, beta> on L^2 (W,T). The second formula is similar to the Feldman-Krupnik-Marcus formula (cf.Gohberg-Krupnik's books). The third formular is very different from their formula. If W is a constant, thenphi becomes a constant, and hence our results become more simple as the following. Let m=(|alpha|^2+|beta|^2)/2, n=||alpha|^2-|beta|^2|/2. Then m+n=max{|alpha|^2, |beta|^2} and m-n=min{|alpha|^2, |beta|^2}. Let ||f|| denote the L^2(T) norm of f. Let H^*(T) denote the Hardy space. Then the following equalities hold. sup{||S_<alpha, beta>f||^2/||f||^2 ; f*L^2(T)}=inf{ess.sup_T(m+(|alphabeta^^_-k|^2+n^2)^<1/2>) ; k*H^*(T)}. inf{||S_<alpha, beta>f||^2/||f||^2 ; f*L^2(T)}=sup{ess.inf_T(m-(|alphabeta^^_-k|^2+n^2)^<1/2>) ; k*H^*(T)}.