Transfer of Fourier Multipliers into Schur Multipliers and Sumsets in a Discrete Group

Neuwirth, Stefan; Ricard, Éric

doi:10.4153/cjm-2011-053-9

Cited by 37 publications

(48 citation statements)

References 26 publications

(37 reference statements)

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“…The L p -theory was not seriously considered until [25]. However, only during very recent years a prolific series of results have appeared in the literature [7,31,32,33,36,40,42].…”

Section: Introductionmentioning

confidence: 99%

Noncommutative De Leeuw Theorems

et al. 2015

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Let H be a subgroup of some locally compact group G. Assume that H is approximable by discrete subgroups and that G admits neighborhood bases which are almost invariant under conjugation by finite subsets of H. Let m : G → C be a bounded continuous symbol giving rise to an L p -bounded Fourier multiplier (not necessarily completely bounded) on the group von Neumann algebra of G for some 1 p ∞. Then, m |H yields an L p -bounded Fourier multiplier on the group von Neumann algebra of H provided that the modular function ∆ G is equal to 1 over H. This is a noncommutative form of de Leeuw's restriction theorem for a large class of pairs (G, H). Our assumptions on H are quite natural, and they recover the classical result. The main difference with de Leeuw's original proof is that we replace dilations of Gaussians by other approximations of the identity for which certain new estimates on almost-multiplicative maps are crucial. Compactification via lattice approximation and periodization theorems are also investigated.

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“…The L p -theory was not seriously considered until [25]. However, only during very recent years a prolific series of results have appeared in the literature [7,31,32,33,36,40,42].…”

Section: Introductionmentioning

confidence: 99%

Noncommutative De Leeuw Theorems

et al. 2015

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“…Fourier multipliers on L p (T d θ ) to Sobolev spaces. Inspired by Neuwirth and Ricard's transference theorem[48], we will relate Fourier multipliers with Schur multipliers. Given a distribution x on T d θ , we write its matrix in the basis (U m ) m∈Z d :[x] = xU n , U m m,n∈Z d = x(m − n)e inθ(m−n) t m,n∈Z d .Here k t denotes the transpose of k = (k 1 , .…”

mentioning

confidence: 99%

Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori

Xiong

Yin

2018

Memoirs of the AMS

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This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative d-torus T d θ (with θ a skew symmetric real d × d-matrix). These spaces share many properties with their classical counterparts. We prove, among other basic properties, the lifting theorem for all these spaces and a Poincaré type inequality for Sobolev spaces. We also show that the Sobolev space W k ∞ (T d θ ) coincides with the Lipschitz space of order k, already studied by Weaver in the case k = 1. We establish the embedding inequalities of all these spaces, including the Besov and Sobolev embedding theorems. We obtain Littlewood-Paley type characterizations for Besov and Triebel-Lizorkin spaces in a general way, as well as the concrete ones in terms of the Poisson, heat semigroups and differences. Some of them are new even in the commutative case, for instance, our Poisson semigroup characterizations improve the classical ones. As a consequence of the characterization of the Besov spaces by differences, we extend to the quantum setting the recent results of Bourgain-Brézis -Mironescu and Maz'ya-Shaposhnikova on the limits of Besov norms. The same characterization implies that the Besov space B α ∞,∞ (T d θ ) for α > 0 is the quantum analogue of the usual Zygmund class of order α. We investigate the interpolation of all these spaces, in particular, determine explicitly the K-functional of the couple (Lp(T d θ ), W k p (T d θ )), which is the quantum analogue of a classical result due to Johnen and Scherer. Finally, we show that the completely bounded Fourier multipliers on all these spaces do not depend on the matrix θ, so coincide with those on the corresponding spaces on the usual d-torus. We also give a quite simple description of (completely) bounded Fourier multipliers on the Besov spaces in terms of their behavior on the Lp-components in the Littlewood-Paley decomposition.

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“…In [39], a transference method has been introduced to overcome the full noncommutativity of quantum tori and to use methods of operator-valued harmonic analysis. Let N θ = L ∞ (T d )⊗T d θ , equipped with the tensor trace ν = dz⊗τ .…”

Section: 2mentioning

confidence: 99%

“…At that time, due to the fact that very little had been done about the analytic aspect, the work of Connes and his collaborators did not include L p -estimates for parametrices and error terms. Recently, inspired by the development on noncommutative harmonic analysis, a lot of progress has been made on Fourier multiplier theory and Calderón-Zygmund theory on noncommutative L p spaces, thanks the efforts of many researchers [33,40,39,7,18,30,61,63]. But so far, the mapping properties of pseudo-differential operators are rarely studied.…”

Section: Introductionmentioning

confidence: 99%

Mapping properties of operator-valued pseudo-differential operators

Xia

Xiong

2019

Journal of Functional Analysis

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In this paper, we investigate the mapping properties of pseudo-differential operators with operator-valued symbols. Thanks to the smooth atomic decomposition of the operatorvalued Triebel-Lizorkin spaces F α,c 1 (R d , M) obtained in our previous paper, we establish the F α,c 1 -regularity of regular symbols for every α ∈ R, and the F α,c 1 -regularity of forbidden symbols for α > 0. As applications, we obtain the same results on the usual and quantum tori.

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Transfer of Fourier Multipliers into Schur Multipliers and Sumsets in a Discrete Group

Cited by 37 publications

References 26 publications

Noncommutative De Leeuw Theorems

Noncommutative De Leeuw Theorems

Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori

Mapping properties of operator-valued pseudo-differential operators

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