Triple Hilbert Transforms Along Polynomial Surfaces

Cho, Yong Kum; Hong, Sungeon; Kim, Joonil; Yang, Chan Woo

doi:10.1007/s00020-009-1731-9

Cited by 5 publications

(4 citation statements)

References 7 publications

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“…So when n = 3, matters are much more delicate but nonetheless a characterisation of L 2 boundedness was found in [3] and depends not only on the support of P but also on the parity of the coefficients. See also [5] for other results in n = 3.…”

Section: Introductionmentioning

confidence: 94%

A class of multiparameter oscillatory singular integral operators: endpoint Hardy space bounds

et al. 2019

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We establish endpoint bounds on a Hardy space H 1 for a natural class of multiparameter singular integral operators which do not decay away from the support of rectangular atoms. Hence the usual argument via a Journétype covering lemma to deduce bounds on product H 1 is not valid.We consider the class of multiparameter oscillatory singular integral operators given by convolution with the classical multiple Hilbert transform kernel modulated by a general polynomial oscillation. Various characterisations are known which give L 2 (or more generally L p , 1 < p < ∞) bounds. Here we initiate an investigation of endpoint bounds on the rectangular Hardy space H 1 in two dimensions; we give a characterisation when bounds hold which are uniform over a given subspace of polynomials and somewhat surprisingly, we discover that the Hardy space and L p theories for these operators are very different.

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Section: Introductionmentioning

confidence: 94%

A class of multiparameter oscillatory singular integral operators: endpoint Hardy space bounds

et al. 2019

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“…Cho, H. Hong, C.W. Yang and the author proved the theorem without assuming the three hypotheses H1-H3 so that Theorem 3.6 (Triple Hilbert transform [4]).…”

Section: S Patelmentioning

confidence: 99%

“…See Figure 3, where normal vectors q ν j are written without the superscript ν = 1 for simplicity. All the faces of N(Λ ν ) for ν = 1, 2 are written as (1,4), (2,5), (3,4), (3,5), (4,5) ,…”

Section: 1mentioning

confidence: 99%

Multiple Hilbert transforms associated with polynomials

Kim¹

2015

Memoirs of the AMS

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Let Λ = (Λ1, · · · , Λ d ) with Λν ⊂ Z n + , and set PΛ the family of all vector polynomials,Given PΛ ∈ PΛ, we consider a class of multi-parameter oscillatory singular integrals,When n = 1, the integral I(PΛ, ξ, r) for any PΛ ∈ PΛ is bounded uniformly in ξ and r.However, when n ≥ 2, the uniform boundedness depends on each indivisual polynomial PΛ. In this paper, we fix Λ and find a necessary and sufficient condition on Λ such that for all PΛ ∈ PΛ, supThe condition is described by faces and their cones of polyhedrons associated with Λν 's.2000 Mathematics Subject Classification. Primary 42B20, 42B25.

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“…More recently, the multiple Hilbert transforms were studied in [6], [8] and [16]. For each (N 1 , N 2 ) ∈ N 2 , consider the lattice point set:…”

Section: Introductionmentioning

confidence: 99%

Discrete Double Hilbert Transforms Along Polynomial Surfaces

Kim¹,

Song²

2022

Preprint

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Triple Hilbert Transforms Along Polynomial Surfaces

Abstract: Given Ω ⊂ Z 3 + , we discuss a necessary and sufficient condition that the triple Hilbert transform associated with any polynomial of the form (t1, t2, t3, m ∈Ω am t m ) is bounded in L p (R 4 ).

Cited by 5 publications

References 7 publications

A class of multiparameter oscillatory singular integral operators: endpoint Hardy space bounds

A class of multiparameter oscillatory singular integral operators: endpoint Hardy space bounds

Multiple Hilbert transforms associated with polynomials

Discrete Double Hilbert Transforms Along Polynomial Surfaces

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