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Journal of Operator Theory

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

Norms of some singular integral operators on weighted L2 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 α and β 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 L2(W) on T. The singular integral operator Sα,β is defined by Sα,βf=αP+f+βPf, fL2(W), where P=IP+. Let h be an outer function such that W=|h|2, and let ϕ be a unimodular function such that ϕ=ˉh/h. In this paper, the norm of Sα,β on L2(W) is calculated in general, using α,β and ϕ. Moreover, if α and β are constant functions, then we give another proof of the Feldman-Krupnik-Markus theorem. If αˉβ belongs to the Hardy space H, we give the theorem which is similar to the Feldman-Krupnik-Markus theorem.


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