Local and multilinear noncommutative de Leeuw theorems

Caspers, Martijn; Janssens, Bas; Krishnaswamy-Usha, Amudhan; Miaskiwskyi, Lukas

doi:10.48550/arxiv.2201.10400

Cited by 2 publications

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“…The same result was then obtained for 𝐺 a locally compact group in [CaSa15]. An analogous result was obtained for actions and crossed products by González-Perez [Gon18] and in an ad hoc way in the bilinear discrete setting a similar result was obtained for the discrete Heisenberg group in [CJKM,Section 7]. The purpose of this paper is to prove transference results for Fourier and Schur multipliers in the multilinear setting for arbitrary unimodular locally compact groups.…”

Section: Caspers Krishnaswamy-usha and Vossupporting

confidence: 65%

Multilinear transference of Fourier and Schur multipliers acting on noncommutative -spaces

Caspers¹,

Krishnaswamy-Usha²,

Vos³

2022

Can. J. Math.-J. Can. Math.

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Let 𝐺 be a locally compact unimodular group, and let 𝜙 be some function of 𝑛 variables on 𝐺. To such a 𝜙, one can associate a multilinear Fourier multiplier, which acts on some 𝑛-fold product of the non-commutative 𝐿 𝑝 -spaces of the group von Neumann algebra. One may also define an associated Schur multiplier, which acts on an 𝑛-fold product of Schatten classes 𝑆 𝑝 (𝐿 2 (𝐺) ). We generalize well-known transference results from the linear case to the multilinear case. In particular, we show that the so-called 'multiplicatively bounded ( 𝑝 1 , . . . , 𝑝 𝑛 )-norm' of a multilinear Schur multiplier is bounded above by the corresponding multiplicatively bounded norm of the Fourier multiplier, with equality whenever the group is amenable. Further, we prove that the bilinear Hilbert transform is not bounded as a vector valued map 𝐿 𝑝 1 (R, 𝑆 𝑝 1 ) × 𝐿 𝑝 2 (R, 𝑆 𝑝 2 ) → 𝐿 1 (R, 𝑆 1 ), whenever 𝑝 1 and 𝑝 2 are such that 1A similar result holds for certain Calderón-Zygmund type operators. This is in contrast to the non-vector valued Euclidean case.

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Section: Caspers Krishnaswamy-usha and Vossupporting

confidence: 65%

Multilinear transference of Fourier and Schur multipliers acting on noncommutative -spaces

Caspers¹,

Krishnaswamy-Usha²,

Vos³

2022

Can. J. Math.-J. Can. Math.

Self Cite

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“…. , t n ) against n j=1 ϕ k (t j ), we can show that lim k lim sup α C k,α = 0, just as in [CJKM,Lemma 4.6].…”

Section: Transference From Fourier To Schur Multiplierssupporting

confidence: 52%

“…We now claim that lim k lim sup α of this expression yields 0, by almost identical arguments as those used in [CJKM,Lemma 4.6]. Since we have a couple of differences, namely that we have a translated function φ k (r 0 • r −1 1 , .…”

Section: Transference From Fourier To Schur Multipliersmentioning

confidence: 71%

“…To show that lim k lim α B k,α = 0, we only note the modifications from [CJKM, Lemma 4.6]. As before, x j = 1 in the proof of [CJKM,Lemma 4.6]. The operators T j appearing in that proof are to be replaced with…”

Section: Transference From Fourier To Schur Multipliersmentioning

confidence: 99%

“…This is not the case with the above definition. As a consequence, we no longer have nice relations between nested linear Fourier multipliers and multilinear Fourier multipliers, as in [CJKM,Lemma 4.4]. As the proof of the multilinear restriction theorem in [CJKM] repeatedly uses such formulae, it is also unclear if these de Leeuw type theorems are still valid in the non-unimodular case.…”

Section: Amenable Groups: Transference From Schur To Fourier Multipliersmentioning

confidence: 99%

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Multilinear transference of Fourier and Schur multipliers acting on non-commutative $L_p$-spaces

Capsers,

Krishnaswamy-Usha,

Vos

2022

Preprint

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Let G be a locally compact unimodular group, and let φ be some function of n variables on G. To such a φ, one can associate a multilinear Fourier multiplier, which acts on some n-fold product of the non-commutative Lp-spaces of the group von Neumann algebra. One may also define an associated Schur multiplier, which acts on an n-fold product of Schatten classes Sp(L2(G)). We generalize well-known transference results from the linear case to the multilinear case. In particular, we show that the so-called 'multiplicatively bounded (p1, . . . , pn)-norm' of a multilinear Schur multiplier is bounded above by the corresponding multiplicatively bounded norm of the Fourier multiplier, with equality whenever the group is amenable. Further, we prove that the bilinear Hilbert transform is not bounded as a vector valued map Lp 1 (R, Sp 1 ) × Lp 2 (R, Sp 2 ) → L1(R, S1), whenever p1 and p2 are such that 1 p 1 + 1 p 2 = 1. A similar result holds for certain Calderón-Zygmund type operators. This is in contrast to the non-vector valued Euclidean case.

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Local and multilinear noncommutative de Leeuw theorems

Cited by 2 publications

References 0 publications

Multilinear transference of Fourier and Schur multipliers acting on noncommutative -spaces

Multilinear transference of Fourier and Schur multipliers acting on noncommutative -spaces

Multilinear transference of Fourier and Schur multipliers acting on non-commutative $L_p$-spaces

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