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

设α和β是单位圆T上的有界可测函数，W是T上的正权函数，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Sα，β；gt；由Sα，β&gt；f=αP_fβP_-f，(f*L^2(W，T))定义，其中P_-=I-P_。该算子与Toeplitz算子和Hankel算子有关。首先，我们研究了Sα，β和gt；在L^2(W，T)上的有界性。由Kooss定理推出，存在许多α，β，W使得P_1是无界的，而Sα，β，W是有界的。接下来，我们研究了Sα、β和gt；的范数。设h是使W=|h|^2的外函数，Phi是使phi=h^_/h的么正函数.利用Hilbert空间变元和Cotlar-Sadosky提升定理，我们得到了用α、β和Phi计算L^2(W，T)上的S的范数的三个公式.第一个公式给出了以L^2(W，T)上的Sα，β；gt；的范数为不动点的实变量的凸函数。第二个公式类似于Feldman-Krupnik-Marcus公式(参看Gohberg-Krupnik的书)。第三个公式与他们的公式非常不同。如果W是一个常量，那么enphi就变成一个常量，因此我们的结果变得更加简单，如下所示。设m=(|Alpha|^2|beta|^2)/2，n=||Alpha|^2-|beta|^2|/2。则m n=max{|Alpha|^2，|beta|^2}，m-n=min{|Alpha|^2，|beta|^2}。设||f||表示f的L^2(T)范数，H^*(T)表示Hardy空间。那么，下列等式成立。Sup{||S_&lt；α，β&>f||^2/||f|^2；f*L^2(T)}=inf{ess.sup_T(m(|Alphabeta^^-k|^2n^2)^&lt；1/2&gt；)；k*H^*(T)}。Inf{||S_&lt；α，β&gt；f||^2/||f||^2；f*L^2(T)}=sup{ess.inf_T(m-(|Alphabeta^_-k|^2n^2)^&lt；1/2&gt；)；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)}.