Weighted Norm Inequalities for Some Singular Integral Operators

Nakazi, Takahiko; Yamamoto, Takanori

doi:10.1006/jfan.1997.3100

Cited by 3 publications

(3 citation statements)

References 11 publications

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“…If r −1 ∈ L ∞ , then (HS)(r) ⊂ (HS). In [9], we defined the another Helson-Szegő type set which is similar to HS(r). We use HS(r) to study the majorization of singular integral operators.…”

Section: Cos V}mentioning

confidence: 99%

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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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Section: Cos V}mentioning

confidence: 99%

“…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. In [10] and [11], for α, β ∈ L ∞ , the norm formula of αP + βQ on the weighted L 2 space 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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“…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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Properties of singular integral operators $$\varvec{S}_{\varvec{\alpha ,\beta }}$$ S α , β

Samanta

Sarkar

2018

Proc Math Sci

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Weighted Norm Inequalities for Some Singular Integral Operators

Cited by 3 publications

References 11 publications

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

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

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

Properties of singular integral operators $$\varvec{S}_{\varvec{\alpha ,\beta }}$$ S α , β

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