Multipliers on Vector-valued L1-spaces for Hypergroups

Sarma, Ritumoni; Kumar, N. Shravan; Kumar, Vishvesh

doi:10.1007/s10114-018-7303-7

Cited by 7 publications

(6 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A Fourier multiplier commutes with the translation operators. In fact, the Fourier multipliers can be characterized using translation operators [46,13,38].…”

Section: 2mentioning

confidence: 99%

“…(46)For a fixed constant c ≥ 1, let us introduce the setR c := u ∈ L ∞ (0, T ; L 2 (R + , Adx)) : u 2 L ∞ (0,T ;L 2 (R + ,Adx)) ≤ cT γ 0 u 0 R + ,Adx) ,with γ 0 > 0 is to be defined later. Now, note thatu 0 (R + ,Adx) + T 3−2γ u 2p L ∞ (0,T ;L 2 (R + ,Adx)) ≤ u 0 (R + ,Adx) + T 3−2γ+γ 0 p c p u 0 2p L 2 (R + ,Adx) .To guarantee u ∈ R c , by invoking(46) we require thatu 0 R + ,Adx) + T 3−2γ+γ 0 p c p u 0 2p L 2 (R + ,Adx) ≤ cT γ 0 u 0 R + ,Adx) .Now by choosing 0 < γ 0 < 2γ−such that γ := 3 − 2γ + γ 0 p < 0, we obtain c R + ,Adx) ≤ cT −γ+γ 0 .…”

mentioning

confidence: 99%

See 1 more Smart Citation

$L^p$-$L^q$ Multipliers on commutative hypergroups

Kumar,

Ruzhansky

2021

Preprint

Self Cite

View full text Add to dashboard Cite

The main purpose of this paper is to prove Hörmander's L p -L q boundedness of Fourier multipliers on commutative hypergroups. We carry out this objective by establishing Paley inequality and Hausdorff-Young-Paley inequality for commutative hypergroups. We show the L p -L q boundedness of the spectral multipliers for the generalised radial Laplacian by examining our results on Chébli-Trimèche hypergroups. As a consequence, we obtain embedding theorems and time asymptotics for the L p -L q norms of the heat kernel for generalised radial Laplacian. Finally, we present applications of the obtained results to study the well-posedness of nonlinear partial differential equations.

show abstract

“…A Fourier multiplier commutes with the translation operators. In fact, the Fourier multipliers can be characterized using translation operators [46,13,38].…”

Section: 2mentioning

confidence: 99%

mentioning

confidence: 99%

$L^p$-$L^q$ Multipliers on commutative hypergroups

Kumar,

Ruzhansky

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…An operator A which is invariant under the left translations will be called a left Fourier multiplier. The left invariant operators can be characterized using the Fourier transform [45,43]. Indeed, if A is a left Fourier multiplier then there exists a function σ A : K → C dπ×dπ , known as the symbol associated with A, such that…”

Section: 3mentioning

confidence: 99%

“…Now, observe that the set {π ∈ K : σ A (π) op ≥ s} is empty for s > A L 2 (K)→L 2 (K) in view of (43) and, therefore, we have…”

Section: 3mentioning

confidence: 99%

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

Kumar,

Ruzhansky

2020

Preprint

Self Cite

View full text Add to dashboard Cite

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

show abstract

“…Study of vector-valued functions has been an active area of research for the past few decades as they provide new ideas in understanding various problems related to partial differential equations and stochastic processes. As a result, several researchers have extended the classical results to the vector-valued setting -Hausdorff-Young inequality [5,17,18,26,24,22], Hardy's inequality [3], singular integrals [6,27], Fourier multipliers [19,33,31] and partial differential equations [1,36].…”

Section: Introductionmentioning

confidence: 99%

Weyl transform of vector-valued functions and vector measures on the phase space associated to a locally compact abelian group

Singhal¹,

Kumar²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we introduce and study the Weyl transform of functions which are integrable with respect to a vector measure on a phase space associated to a locally compact abelian group. We also study the Weyl transform of vector measures. Later, we also introduce and study the convolution of functions from L p -spaces associated to a vector measure. We also study the Weyl transform of vector-valued functions and prove a vector-valued analogue of the Hausdorff-Young inequality.

show abstract

Multipliers on Vector-valued L1-spaces for Hypergroups

Cited by 7 publications

References 15 publications

$L^p$-$L^q$ Multipliers on commutative hypergroups

$L^p$-$L^q$ Multipliers on commutative hypergroups

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

Weyl transform of vector-valued functions and vector measures on the phase space associated to a locally compact abelian group

Contact Info

Product

Resources

About