Function spaces and classes of pseudodifferential operators
Wojciech Czaja and Ziemowit Rzeszotnik
Source: Tohoku Math. J. (2) Volume 55, Number 1 (2003), 131-140.
Abstract
We introduce a new class of selfadjoint compact pseudodifferential operators, which is analogous to a class of elliptic unbounded pseudodifferential operators and is, therefore, suitable for obtaining upper and lower estimates on the eigenvalues of operators in this class. We prove such estimates and, as an application, we show that any operator from this class belongs to the Schatten-von Neuman class if and only if its symbol belongs to the Lorentz space.
Primary Subjects: 35S05
Secondary Subjects: 46E10, 46F05, 47G30
Keywords: Lorentz space; pseudodifferential operators; Schatten class; spectral asymptotics
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.tmj/1113247450
Mathematical Reviews number (MathSciNet):
MR1956085
Zentralblatt MATH identifier:
1051.47038
Digital Object Identifier: doi:10.2748/tmj/1113247450
References
R. Beals, Characterization of pseudodifferential operators and applications, Duke Math. J. 44 (1977), 45--57.
Mathematical Reviews (MathSciNet):
MR435933
Digital Object Identifier: doi:10.1215/S0012-7094-77-04402-7
Project Euclid: euclid.dmj/1077312091
W. Czaja and Z. Rzeszotnik, Two remarks about spectral asymptotics of pseudodifferential operators, Colloq. Math. 80 (1999), 131--145.
Mathematical Reviews (MathSciNet):
MR1684577
W. Czaja and Z. Rzeszotnik, Pseudodifferential operators and Gabor frames: spectral asymptotics, Math. Nachr. 233/234 (2002), 77--88.
Mathematical Reviews (MathSciNet):
MR1879864
Digital Object Identifier: doi:10.1002/1522-2616(200201)233:1<77::AID-MANA77>3.3.CO;2-2
H. G. Feichtinger and T. Strohmer, Gabor analysis and algorithms, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 1998.
Mathematical Reviews (MathSciNet):
MR1601119
G. B. Folland, Harmonic analysis in phase space, Princeton University Press, Princeton, NJ, 1989.
Mathematical Reviews (MathSciNet):
MR983366
Zentralblatt MATH:
0682.43001
P. Głowacki, The Weyl asymptotic formula for a class of pseudodifferential operators, Studia Math. 127 (1998), 169--190.
Mathematical Reviews (MathSciNet):
MR1488149
K. Gröchenig, Foundations of time--frequency analysis, Birkhäuser Boston, Boston, MA, 2001.
Mathematical Reviews (MathSciNet):
MR1843717
K. Gröchenig, personal communication.
K. Gröchenig and C. Heil, Modulation spaces and pseudodifferential, operators Integral Equations<br/> Operator Theory 34 (1999), 439--457.
Mathematical Reviews (MathSciNet):
MR1702232
Digital Object Identifier: doi:10.1007/BF01272884
C. Heil, J. Ramanathan and P. Topiwala, Singular values of compact pseudodifferential operators, J. Funct. Anal. 150 (1997), 426--452.
Mathematical Reviews (MathSciNet):
MR1479546
Digital Object Identifier: doi:10.1006/jfan.1997.3127
R. Howe, Quantum mechanics and partial differential equations, J. Funct. Anal. 38 (1980), 188--254.
Mathematical Reviews (MathSciNet):
MR587908
Digital Object Identifier: doi:10.1016/0022-1236(80)90064-6
L. Hörmander, On the asymptotic distribution of the pseudodifferential operators in $\br ^n$, Ark. Mat. 17 (1979), 297--313.
Mathematical Reviews (MathSciNet):
MR608322
Digital Object Identifier: doi:10.1007/BF02385475
L. Hörmander, The analysis of linear partial differential operators, Springer, Berlin, 1983.
Mathematical Reviews (MathSciNet):
MR404822
J. C. T. Pool, Mathematical aspects of the Weyl correspondence, J. Math. Phys. 7 (1966), 66--76.
Mathematical Reviews (MathSciNet):
MR204049
Digital Object Identifier: doi:10.1063/1.1704817
R. Rochberg and K. Tachizawa, Pseudodifferential operators, Gabor frames and local trigonometric bases, Gabor analysis and algorithms, 171--192, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 1998.
Mathematical Reviews (MathSciNet):
MR1601103
Zentralblatt MATH:
0890.42009
C. Rondeaux, Classes de Schatten d'opérateurs pseudo-différentiels, Ann. Sci. École Norm. Sup. (4) 17 (1984), 67--81.
Mathematical Reviews (MathSciNet):
MR744068
W. Stenger and A. Weinstein, Methods of intermediate problems for eigenvalues, Mathematics in Science and Engineering, Vol. 89, Academic Press, New York-London, 1972.
Mathematical Reviews (MathSciNet):
MR477971
Zentralblatt MATH:
0291.49034
K. Tachizawa, The pseudodifferential operators and Wilson bases, J. Math. Pures Appl. (9) 75 (1996), 509--529.
Mathematical Reviews (MathSciNet):
MR1423045
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