Multipliers of Sobolev spaces

Poornima, S.

doi:10.1016/0022-1236(82)90002-7

Cited by 12 publications

(5 citation statements)

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“…However, this is no longer the case as soon as d ≥ 2, as proved by Poornima [59] in the commutative case for R d . Poornima's example comes from Ornstein [50] which is still valid for our setting.…”

Section: Embedding Of Sobolev Spacesmentioning

confidence: 92%

“…We now investigate Fourier multipliers on Sobolev spaces. We refer to [59,9] for the study of Fourier multipliers on the classical Sobolev spaces. If X is a Banach space of distributions on T d θ , we denote by M(X) the space of bounded Fourier multipliers on X; if X is further equipped with an operator space structure, M cb (X) is the space of c.b.…”

Section: Embedding Of Sobolev Spacesmentioning

confidence: 99%

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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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Section: Embedding Of Sobolev Spacesmentioning

confidence: 92%

Section: Embedding Of Sobolev Spacesmentioning

confidence: 99%

Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori

Xiong

Yin

2018

Memoirs of the AMS

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“…The Laguerre Sobolev spaces are defined in terms of certain vector fields associated to L, see subsection 2.4 for details. Analogous to the work of Poornima [13], it was shown in [14] that for 1 < p < ∞, the space of Weyl multipliers of any Laguerre Sobolev space W N,p L (C n ) coincides with that of L p (C n ) with norm equivalence. They also characterised the space of Weyl multipliers of W N,1 L (C n ), showing it to be the dual of certain function space.…”

Section: Introduction and The Main Resultsmentioning

confidence: 72%

“…Given a Fourier multiplier T m on L p (R n ), 1 ≤ p ≤ ∞ it is natural to ask if it is also a bounded linear operator on the Sobolev space W N,p (R n ) consisting of all f ∈ L p (R n ) whose distributional derivatives ∂ α f also belong to L p (R n ) for all |α| ≤ N. In [13], it was shown that for 1 < p < ∞, the Fourier multiplier space of any Sobolev space W N,p (R n ) is the same as that of L p (R n ) with norm equivalence, and that the analogous result for p = 1 is true only for functions of one real-variable. They have established that the Fourier multiplier space of W N,1 (R n ) is strictly larger than that of L 1 (R n ) when n ≥ 2.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Homogeneous Fourier and Weyl multipliers on Sobolev spaces related to the Heisenberg group

Basak,

Garg,

Thangavelu

2020

Preprint

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“…Fourier multipliers for Sobolev spaces W k,p (R n ), k ∈ N, 1 ≤ p < ∞ have been characterized by Poornima in [5]. Weyl multipliers for Laguerre Sobolev spaces W k,p L (C n ) and Hermite multipliers for Hermite Sobolev spaces W k,p H (R n ) were studied by Radha and Thangavelu in [7] and [8].…”

Section: Notations and Backgroundmentioning

confidence: 99%

Fourier multipliers for Sobolev spaces on the Heisenberg group

2010

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A b s t r a c t . In this paper, it is shown that the class of right Fourier multipliers for the Sobolev space W k,p (H n ) coincides with the class of right Fourier multipliers forand L is the sublaplacian on H n . This proof is based on the Calderon-Zygmund theory on the Heisenberg group. It is also shown that when p = 1, the class of right multipliers for the Sobolev space W k,1 (H n ) coincides with the dual space of the projective tensor product of two function spaces.

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Multipliers of Sobolev spaces

Cited by 12 publications

References 5 publications

Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori

Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori

Homogeneous Fourier and Weyl multipliers on Sobolev spaces related to the Heisenberg group

Fourier multipliers for Sobolev spaces on the Heisenberg group

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