Journal of Operator Theory

Volume 50, Issue 2, Fall 2003 pp. 311-330.

Norms of some singular integral operators on weighted $L^2$ space

Authors: Takahiko Nakazi (1) and Takanori Yamamoto (2)
Author institution: (1) Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan
(2) Department of Mathematics, Hokkai-Gakuen University, Sapporo 062-8605, Japan

Summary: Let $\alpha$ and $\beta$ be measurable functions on the unit circle $T$, and let $W$ be a positive function on $T$ such that the Riesz projection $P_+$ is bounded on the weighted space $L^2(W)$ on $T$. The singular integral operator $S_{\alpha,\beta}$ is defined by $S_{\alpha,\beta} f = \alpha P_+ f + \beta P_- f $, $ f \in L^2(W)$, where $P_- = I - P_+$. Let $h$ be an outer function such that $W = |h|^2$, and let $\phi$ be a unimodular function such that $\phi = \bar{h}/h$. In this paper, the norm of $S_{\alpha,\beta}$ on $L^2(W)$ is calculated in general, using $\alpha, \beta$ and $\phi$. Moreover, if $\alpha$ and $\beta$ are constant functions, then we give another proof of the Feldman-Krupnik-Markus theorem. If $\alpha\bar{\beta}$ belongs to the Hardy space $H^\infty$, we give the theorem which is similar to the Feldman-Krupnik-Markus theorem.

Keywords: Singular integral operator, norm, Hardy space, Helson-Szeg\"{o} weight, {\rm (A$_2$)}-condition

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