The Hausdorff–Young inequality for Orlicz spaces on compact hypergroups

Kumar, Vishvesh; Sarma, Ritumoni

doi:10.4064/cm7627-4-2019

Cited by 9 publications

(8 citation statements)

References 13 publications

(27 reference statements)

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“…The Hausdorff-Young inequality is a classical result for the Fourier analysis on groups. Recent proofs of the inequality for DJS hypergroups appear in [DS13] for commutative hypergroups and in [KS20], [KR20].…”

Section: P Spaces and Fourier Inequalitiesmentioning

confidence: 99%

Galois Correspondence and Fourier Analysis on Local Discrete Subfactors

Bischoff,

Del Vecchio,

Giorgetti

2021

Preprint

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Discrete subfactors include a particular class of infinite index subfactors and all finite index ones. A discrete subfactor is called local when it is braided and it fulfills a commutativity condition motivated by the study of inclusion of Quantum Field Theories in the algebraic Haag-Kastler setting. In [BDVG21], we proved that every irreducible local discrete subfactor arises as the fixed point subfactor under the action of a canonical compact hypergroup. In this work, we prove a Galois correspondence between intermediate von Neumann algebras and closed subhypergroups, and we study the subfactor theoretical Fourier transform in this context. Along the way, we extend the main results concerning α-induction and σ-restriction for braided subfactors previously known in the finite index case. Contents1. Introduction 2. Preliminaries 2.1. The canonical endomorphism 2.2. Conditional expectations 2.3. Braided and local subfactors 2.4. Compact hypergroups 2.5. Duality theorem and dominated UCP maps 3. α-induction for discrete subfactors 4. Intermediate inclusions 5. Galois correspondence 6. Fourier transform 6.1. Extension of the Fourier transform to UCP maps 6.2. The local discrete case: Fourier transform on measures 6.3. L p spaces and Fourier inequalities 6.4. Involutions, convolutions and products 6.5. Convolution inequalities 6.6. Inversion formula and uncertainty principles References

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Section: P Spaces and Fourier Inequalitiesmentioning

confidence: 99%

Galois Correspondence and Fourier Analysis on Local Discrete Subfactors

Bischoff,

Del Vecchio,

Giorgetti

2021

Preprint

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“…Recently, the first author with R. Sarma [31] also obtained Hausdorff-Young inequality using different norm which was useful to study Hausdorff-Young inequality for Orlicz spaces [31]. We will discuss it in the next section in more details.…”

Section: Preliminariesmentioning

confidence: 99%

“…The following Hausdorff-Young inequality for Fourier transform on compact hypergroups was recently obtained by the first author and R. Sarma [31].…”

Section: Hausdorff-young-paley and Hardy-littlewood Inequalities On C...mentioning

confidence: 99%

“…In the view of Proposition 3.2 one can see that the Hausdorff-Young inequality (17) using Schatten p-norm is sharper than inequality (30). In [31], Theorem 3.3 is further used to define Orlicz space on dual of compact hypergroups and to obtained Hausdorff-Young inequality for Orlicz spaces on compact hypergroup.…”

Section: Hausdorff-young-paley and Hardy-littlewood Inequalities On C...mentioning

confidence: 99%

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Hardy-Littlewood inequality and $L^p$-$L^q$ Fourier multipliers on compact hypergroups

Kumar,

Ruzhansky

2020

Preprint

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This paper deals with the inequalities devoted to the comparison between the norm of a function on a compact hypergroup and the norm of its Fourier coefficients.We prove the classical Paley inequality in the setting of compact hypergroups which further gives the Hardy-Littlewood and Hausdorff-Young-Paley (Pitt) inequalities in the noncommutative context. We establish Hörmander's L p -L q Fourier multiplier theorem on compact hypergroups for 1 < p ≤ 2 ≤ q < ∞ as an application of Hausdorff-Young-Paley inequality. We examine our results for the hypergroups constructed from the conjugacy classes of compact Lie groups and for a class of countable compact hypergroups. Contents 1. Introduction 2. Preliminaries 2.1. Definitions and representations of compact hypergroups 2.2. Fourier analysis on compact hypergroups 2.3. Commutative compact hypergroups 3. Hausdorff-Young-Paley and Hardy-Littlewood inequalities on compact hypergroups 3.1. Paley inequality on compact hypergroups 3.2. Hardy-Littlewood inequality on compact hypergroups 3.3. Hausdorff-Young-Paley inequality on compact hypergroups 4. L p -L q -boundedness of Fourier multipliers on compact hypergroups 5. Examples of hypergroups 5.1. Conjugacy classes of compact Lie groups

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“…M. M. Rao [19] studied the Hausdorff-Young inequality for Orlicz spaces on locally compact abelian groups. In fact, the celebrated work of M. M. Rao in the context of Orlicz spaces on locally compact groups (see [19,21,22,23]) motivated the first author to study the Hausdorff-Young inequality for Orlicz spaces on compact hypergroups [14,13,15]. Recently, the second author with his collaborators established non-commutative version of the aforementioned inequality and of Hardy-Littlewood inequalities on compact homogeneous manifolds and locally compact groups [1,3,2,4].…”

Section: Introductionmentioning

confidence: 99%