Some singular integral operators and Helson-Szegö measures

Nakazi, Takahiko; Yamamoto, Takanori

doi:10.1016/0022-1236(90)90111-w

Cited by 5 publications

(3 citation statements)

References 7 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Samanta,

Sarkar

2015

Preprint

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For α, β ∈ L ∞ (S 1 ), the singular integral operator S α,β on L 2 (S 1 ) is defined by S α,β f := αP f + βQf , where P denotes the orthogonal projection of L 2 (S 1 ) onto the Hardy space H 2 (S 1 ), and Q denotes the orthogonal projection onto H 2 (S 1 ) ⊥ . In a recent paper Nakazi and Yamamoto have studied the normality and self-adjointness of S α,β . This work has shown that S α,β may have analogous properties to that of the Toeplitz operator. In this paper we study several other properties of S α,β .

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Section: Introductionmentioning

confidence: 99%

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

Samanta,

Sarkar

2015

Preprint

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“…Let α, β ∈ L ∞ , and W is a nonnegative function in L 1 . In [8] and [16], the condition of α, β and W such that αP + βQ is contractive was given. In [9], the conditions of α, β and W such that αP + βQ is bounded and bounded below was given.…”

Section: Introductionmentioning

confidence: 99%

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

Yamamoto¹

2014

Ann. Funct. Anal.

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Let a, b, c and d be functions in L 2 = L 2 (T, dθ/2π), where T denotes the unit circle. Let P denote the set of all trigonometric polynomials. Suppose the singular integral operators A and B are defined by A = aP + bQ and B = cP + dQ on P, where P is an analytic projection and Q = I − P is a co-analytic projection. In this paper, we use the Helson-Szegő type set (HS)(r) to establish the condition of a, b, c and d satisfying Af 2 ≥ Bf 2 for all f in P. If a, b, c and d are bounded measurable functions, then A and B are bounded operators, and this is equivalent to that B is majorized by A on L 2 , i.e., A * A ≥ B * B on L 2 . Applications are then presented for the majorization of singular integral operators on weighted L 2 spaces, and for the normal singular integral operators aP + bQ on L 2 when a − b is a complex constant.

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“…Hunt et al [5] showed that HS consists of weights which satisfy the (A 2 ) condition. Nakazi and Yamamoto [8] and Yamamoto [12] studied the first one-weighted inequality for C=1. They gave necessary and sufficient conditions that are more complicated than HS r .…”

Section: Introductionmentioning

confidence: 99%