Extensions of the vector-valued Hausdorff–Young inequalities

Oscar Dominguez; Mark Veraar

doi:10.1007/s00209-020-02675-6

Extensions of the vector-valued Hausdorff–Young inequalities

Oscar Dominguez^*, Mark Veraar

^*Corresponding author for this work

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Abstract

In this paper we study the vector-valued analogues of several inequalities for the Fourier transform. In particular, we consider the inequalities of Hausdorff–Young, Hardy–Littlewood, Paley, Pitt, Bochkarev and Zygmund. The Pitt inequalities include the Hausdorff–Young and Hardy–Littlewood inequalities and state that the Fourier transform is bounded from L^p(R^d, | · | ^β^p) into L^q(R^d, | · | ^-^γ^q) under certain condition on p, q, β and γ. Vector-valued analogues are derived under geometric conditions on the underlying Banach space such as Fourier type and related geometric properties. Similar results are derived for T^d and Z^d by a transference argument. We prove sharpness of our results by providing elementary examples on ℓ^p-spaces. Moreover, connections with Rademacher (co)type are discussed as well.

Original language	English
Pages (from-to)	373-425
Number of pages	53
Journal	Mathematische Zeitschrift
Volume	299
Issue number	1-2
DOIs	https://doi.org/10.1007/s00209-020-02675-6
Publication status	Published - 2021

Bibliographical note

Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care

Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

Keywords

(co)type
Banach space
Bochkarev
Fourier type
Hardy–Littlewood
Hausdorff–Young
Limiting interpolation
Paley
Pitt
Vector-valued Fourier transform
Zygmund

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title = "Extensions of the vector-valued Hausdorff–Young inequalities",

abstract = "In this paper we study the vector-valued analogues of several inequalities for the Fourier transform. In particular, we consider the inequalities of Hausdorff–Young, Hardy–Littlewood, Paley, Pitt, Bochkarev and Zygmund. The Pitt inequalities include the Hausdorff–Young and Hardy–Littlewood inequalities and state that the Fourier transform is bounded from Lp(Rd, | · | βp) into Lq(Rd, | · | -γq) under certain condition on p, q, β and γ. Vector-valued analogues are derived under geometric conditions on the underlying Banach space such as Fourier type and related geometric properties. Similar results are derived for Td and Zd by a transference argument. We prove sharpness of our results by providing elementary examples on ℓp-spaces. Moreover, connections with Rademacher (co)type are discussed as well.",

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doi = "10.1007/s00209-020-02675-6",

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AU - Veraar, Mark

N1 - Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

PY - 2021

Y1 - 2021

N2 - In this paper we study the vector-valued analogues of several inequalities for the Fourier transform. In particular, we consider the inequalities of Hausdorff–Young, Hardy–Littlewood, Paley, Pitt, Bochkarev and Zygmund. The Pitt inequalities include the Hausdorff–Young and Hardy–Littlewood inequalities and state that the Fourier transform is bounded from Lp(Rd, | · | βp) into Lq(Rd, | · | -γq) under certain condition on p, q, β and γ. Vector-valued analogues are derived under geometric conditions on the underlying Banach space such as Fourier type and related geometric properties. Similar results are derived for Td and Zd by a transference argument. We prove sharpness of our results by providing elementary examples on ℓp-spaces. Moreover, connections with Rademacher (co)type are discussed as well.

AB - In this paper we study the vector-valued analogues of several inequalities for the Fourier transform. In particular, we consider the inequalities of Hausdorff–Young, Hardy–Littlewood, Paley, Pitt, Bochkarev and Zygmund. The Pitt inequalities include the Hausdorff–Young and Hardy–Littlewood inequalities and state that the Fourier transform is bounded from Lp(Rd, | · | βp) into Lq(Rd, | · | -γq) under certain condition on p, q, β and γ. Vector-valued analogues are derived under geometric conditions on the underlying Banach space such as Fourier type and related geometric properties. Similar results are derived for Td and Zd by a transference argument. We prove sharpness of our results by providing elementary examples on ℓp-spaces. Moreover, connections with Rademacher (co)type are discussed as well.

KW - (co)type

KW - Banach space

KW - Bochkarev

KW - Fourier type

KW - Hardy–Littlewood

KW - Hausdorff–Young

KW - Limiting interpolation

KW - Paley

KW - Pitt

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EP - 425

JO - Mathematische Zeitschrift

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IS - 1-2

ER -

Extensions of the vector-valued Hausdorff–Young inequalities

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Keywords

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