Norms and Essential Norms of the Singular Integral Operator with Cauchy Kernel on Weighted Lebesgue Spaces

Abstract

Let α and β be bounded measurable functions on the unit circle T, and let L 2 (W ) be a weighted L 2 space on T. The singular inte-where P is an analytic projection and Q = I − P is a co-analytic projection. In the previous paper, the essential norm of S α,β are calculated in the case when W is a constant function. In this paper, the essential norm of S α,β are estimated in the case when W is an A2-weight. Mathematics Subject Classification (2000). Primary 45E10; Secondary 47B35.

Majorization of Singular Integral Operators with Cauchy Kernel on $L^2$

“…αP + βQ W and P W denote the operator norms of each operators on L 2 (W ). In [11], this theorem was generalized to the case when α and β are functions in L ∞ .…”

“…In [9], the conditions of α, β and W such that αP + βQ is bounded and bounded below was given. In [10] and [11], for α, β ∈ L ∞ , the norm formula of αP + βQ on the weighted L 2 space was given. In [10], [11] and [16], the another proofs of Feldman-…”

“…In [10] and [11], for α, β ∈ L ∞ , the norm formula of αP + βQ on the weighted L 2 space was given. In [10], [11] and [16], the another proofs of Feldman-…”

Algebras of Generalized Singular Integral Operators with Cauchy kernel

“…is the singular integral operator. T. Nakazi and T. Yamamoto [21,19,20,22,23,24] have study the boundedness and normality of S α,β , and calculate its norm, C. Gu [16] have study the algebraic properties of S α,β . (4) Toeplitz plus Hankel operators: if H = f 0 g 0 , then (I ⊕ J)R H | H 2 = T f + Γ g , where Jx(z) = zx(z) for x ∈ L 2 (T).…”

Properties of singular integral operators $S_{α,β}$

“…Some properties of S α,β related to the norm (or weighted norm), invertibility, boundedness (with weight) etc have been studied in ( [3], [4] , [5], [6], [7], [8], [9], [10], [11], [12]). Some of these results are generalizations of properties of S. In a recent paper ( [10]) the normality and selfadjointness of the operator S α,β are studied.…”