Trace ideals for pseudo-differential operators and their commutators with symbols in α-modulation spaces

Kobayashi, Masaharu; Sugimoto, M.; Tomita, Naohide

doi:10.1007/s11854-009-0006-3

Cited by 17 publications

(11 citation statements)

References 42 publications

(103 reference statements)

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“…The use of the letter a for the elements of M ∞,1 v0, s (R 2n ) suggests that this space could be used as a symbol class. This is indeed the case; we mention that when s = 0 one recovers the so-called "Sjöstrand classes" [59,60] which have been studied and developed in [5,34,35,47] and [49,[64][65][66] from the point of view of modulation space theory. As a symbol class is rather large; for instance…”

Section: Lemma 1 Let λ Be a Real Number Different From Zero And Setmentioning

confidence: 52%

“…Due to limitations of length and time we have only been able to give a few applications of the theory of modulation spaces to the Landau-Weyl calculus. Modulation spaces and related topics shave turned out to be the proper setting for the discussion of pseudodifferential operators in the last decade, see for instance the papers [32,37] and the references therein; also see [5,28,[47][48][49][64][65][66]. For related topics such as the spaces q (L p ) see for instance the work of Birman and Solomjak [3], Christ and Kiselev [6] or Simon [57].…”

Section: Discussionmentioning

confidence: 99%

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Spectral and regularity properties of a pseudo-differential calculus related to Landau quantization

Gosson

Luef

2010

J. Pseudo-Differ. Oper. Appl.

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The theme of this work is that the theory of charged particles in a uniform magnetic field can be generalized to a large class of operators if one uses an extended a class of Weyl operators which we call "Landau-Weyl pseudodifferential operators". The link between standard Weyl calculus and Landau-Weyl calculus is made explicit by the use of an infinite family of intertwining "windowed wavepacket transforms"; this makes possible the use of the theory of modulation spaces to study various regularity properties. Our techniques allow us not only to recover easily the eigenvalues and eigenfunctions of the Hamiltonian operator of a charged particle in a uniform magnetic field, but also to prove global hypoellipticity results and to study the regularity of the solutions to Schrödinger equations.

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Section: Lemma 1 Let λ Be a Real Number Different From Zero And Setmentioning

confidence: 52%

Section: Discussionmentioning

confidence: 99%

Spectral and regularity properties of a pseudo-differential calculus related to Landau quantization

Gosson

Luef

2010

J. Pseudo-Differ. Oper. Appl.

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“…Also, due to the above mentioned smoothing effect one expects Born-Jordan operators enjoy better continuity properties than those known for Weyl operators (cf. [5,9,10,11,13,19,20,23,30,31,32,33,34,35] and the references therein). Some results in this direction were already obtained in [2,36], and we plan to continue this study in a subsequent work [8].…”

Section: Every Linear Continuous Operatormentioning

confidence: 99%

On the invertibility of Born–Jordan quantization

Cordero

Gosson

Nicola

2016

Journal de Mathématiques Pures et Appliquées

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As a consequence of the Schwartz kernel Theorem, any linear continuous operator A : S(R n ) −→ S ′ (R n ) can be written in Weyl form in a unique way, namely it is the Weyl quantization of a unique symbol a ∈ S ′ (R 2n ). Hence, dequantization can always be performed, and in a unique way. Despite the importance of this topic in Quantum Mechanics and Time-frequency Analysis, the same issue for the Born-Jordan quantization seems simply unexplored, except for the case of polynomial symbols, which we also review in detail. In this paper we show that any operator A as above can be written in Born-Jordan form, although the representation is never unique if one allows general temperate distributions as symbols. Then we consider the same problem when the space of temperate distributions is replaced by the space of smooth slowly increasing functions which extend to entire function in C 2n , with a growth at most exponential in the imaginary directions. We prove again the validity of such a representation, and we determine a sharp threshold for the exponential growth under which the representation is unique. We employ techniques from the theory of division of distributions.2010 Mathematics Subject Classification. 47G30, 81S05, 46F10.

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“…It is known that they include Besov spaces B p,q s and modulation spaces M p,q s as special cases corresponding to α=1 and α=0. As a corollary of Theorem 1.2, we can obtain a more general result in terms of α-modulation spaces which is an extension of the result of L 2 -boundedness in Kobayashi-Sugimoto-Tomita [10].…”

Section: Application Of Theorem 12 To Function Spacesmentioning

confidence: 69%