# Subalgebras to a Wiener type algebra of pseudo-differential operators

Joachim Toft^{[1]}

- [1] Blekinge Technical University, Department of Mathematics, IHN, 371-79 Karlskrona (Suède)

Annales de l’institut Fourier (2001)

- Volume: 51, Issue: 5, page 1347-1383
- ISSN: 0373-0956

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topToft, Joachim. "Subalgebras to a Wiener type algebra of pseudo-differential operators." Annales de l’institut Fourier 51.5 (2001): 1347-1383. <http://eudml.org/doc/115950>.

@article{Toft2001,

abstract = {We study general continuity properties for an increasing family of Banach spaces $S^p_w$ of classes for pseudo-differential symbols, where $S^\infty _w=S_w$ was introduced by J.
Sjöstrand in 1993. We prove that the operators in $\{\rm Op\}(S^p_w)$ are Schatten-von
Neumann operators of order $p$ on $L^2$. We prove also that $\{\rm Op\}(S^p_w)\{\rm Op\}(S^r_w)\subset \{\rm Op\}(S^r_w)$ and $S^p_w\cdot S^q_w\subset S^r_w$, provided $1/p +
1/q =1/r$. If instead $1/p +1/q = 1+1/r$, then $S^p_ww * S^q_w\subset S^r_w$. By
modifying the definition of the $S^p_w$-spaces, one also obtains symbol classes related
to the $S(m,g)$ spaces.},

affiliation = {Blekinge Technical University, Department of Mathematics, IHN, 371-79 Karlskrona (Suède)},

author = {Toft, Joachim},

journal = {Annales de l’institut Fourier},

keywords = {pseudo-differential operators; Weyl calculus; Schatten-von Neumann classes; admissible functions; Hölder's inequality; Young's inequality; subalgebras; properties of pseudodifferential operators; symbols},

language = {eng},

number = {5},

pages = {1347-1383},

publisher = {Association des Annales de l'Institut Fourier},

title = {Subalgebras to a Wiener type algebra of pseudo-differential operators},

url = {http://eudml.org/doc/115950},

volume = {51},

year = {2001},

}

TY - JOUR

AU - Toft, Joachim

TI - Subalgebras to a Wiener type algebra of pseudo-differential operators

JO - Annales de l’institut Fourier

PY - 2001

PB - Association des Annales de l'Institut Fourier

VL - 51

IS - 5

SP - 1347

EP - 1383

AB - We study general continuity properties for an increasing family of Banach spaces $S^p_w$ of classes for pseudo-differential symbols, where $S^\infty _w=S_w$ was introduced by J.
Sjöstrand in 1993. We prove that the operators in ${\rm Op}(S^p_w)$ are Schatten-von
Neumann operators of order $p$ on $L^2$. We prove also that ${\rm Op}(S^p_w){\rm Op}(S^r_w)\subset {\rm Op}(S^r_w)$ and $S^p_w\cdot S^q_w\subset S^r_w$, provided $1/p +
1/q =1/r$. If instead $1/p +1/q = 1+1/r$, then $S^p_ww * S^q_w\subset S^r_w$. By
modifying the definition of the $S^p_w$-spaces, one also obtains symbol classes related
to the $S(m,g)$ spaces.

LA - eng

KW - pseudo-differential operators; Weyl calculus; Schatten-von Neumann classes; admissible functions; Hölder's inequality; Young's inequality; subalgebras; properties of pseudodifferential operators; symbols

UR - http://eudml.org/doc/115950

ER -

## References

top- A. Boulkhemair, Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators, Math. Res. L. 4 (1997), 53-67 Zbl0905.35103MR1432810
- J. Bergh, J. Löfström, Interpolation Spaces. An introduction, (1976), Springer-Verlag, Berlin-Heidelberg-New York Zbl0344.46071MR482275
- M. Dimassi, J. Sjöstrand, Spectral Asymptotics in the Semi-Classical Limit, vol. 268 (1999), Cambridge University Press, Cambridge, New York, Melbourne, Madrid Zbl0926.35002MR1735654
- L. Hörmander, The Analysis of Linear Partial Differential Operators, vol. III (1985), Springer-Verlag, Berlin-Heidelberg-New York-Tokyo Zbl0601.35001MR404822
- E. H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math. 102 (1990), 179-208 Zbl0726.42005MR1069246
- M. Reed, B. Simon, Methods of modern mathematical physics, (1979), Academic Press, London-New York Zbl0405.47007
- B. Simon, Trace ideals and their applications I, vol. 35 (1979), Cambridge University Press, Cambridge-London-New York-Melbourne Zbl0423.47001MR541149
- J. Sjöstrand, An algebra of pseudodifferential operators, Math. Res. L. 1 (1994), 185-192 Zbl0840.35130MR1266757
- J. Sjöstrand, Wiener type algebras of pseudodifferential operators, Séminaire Equations aux Dérivées Partielles, Ecole Polytechnique, 1994 n°IV (1995) Zbl0880.35145
- J. Toft, Continuity and Positivity Problems in Pseudo-Differential Calculus, (1996)
- J. Toft, Regularizations, decompositions and lower bound problems in the Weyl calculus, Comm. Partial Differential Equations 7-8 (2000), 1201-1234 Zbl0963.35215
- J. Toft, Continuity properties in non-commutative convolution algebras with applications in pseudo-differential calculus, Bull. Sci. Math. 125 (2001) Zbl1002.43003MR1906240

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