Abstract
We characterize the spectra of Lp-bounded translation invariant pseudo-differential operators with symbols in the Hörmander class \(S_{\rho }^{m} \). In particular, we obtain for these operators a precise version of their Lp-spectral invariance. We also prove a partial result on the spectra of Lp-bounded pseudodifferential operators with symbols in the Hörmander class \(S_{{\rho ,\delta }}^{m} \). We use these results to study the holomorphic functional calculus of translation invariant pseudo-differentia1 operators.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
J. Alvarez, A. P. Calderón: Functional calculi for pseudo-differential operators, I, Proc. Sem. Fourier Anal., El Escorial, Spain, (Editors: M. de Guzmán, I. Peral) (1979), 1–61.
J. Alvarez, J. Hounie: Estimates for the kernel and continuity properties of pseudo-differential operators, Arkiv för Mat. 28 (1990), 1–22.
J. Alvarez, J. Hounie: Spectral invariance and tameness of pseudo- differential operators on weighted Sobolev spaces, J. of Op. Th. 30 (1993), 41–67.
R. Beals: Characterization of pseudo-differential operators and applications, Duke Math. J. 44 (1977), 45–57.
C. Fefferman, LP bounds for pseudo-differential operators, Israel J. Math. 14 (1973), 413–417.
L. Hörmander: Pseudo-differential operators and hypoelliptic equations, Proc. Symp. Pure Math. 10, Amer. Math. Soc. (1967), 138–183.
S. Igari: Functions of L p-multipliers, Tôhoku Math. J. 21 (1969), 304–320.
H. Kumano-go: Oscillatory integrals of symbols of pseudo-differential operators and the local solvability theorem of Nirenberg and Trèves, Kakata Symp. on PDE (1972), 166–191.
F. Riesz, B. Sz-Nagy: Functional analysis, Dover (1990).
P. Sarnak: Spectra of singular measures as multipliers on L P, J. of Funct. Anal. 37 (1980), 302–317.
E. Schrohe: Boundedness and spectral invariance for standard pseudo- differential operators on anisotropically weighted L P-Sobolev spaces, Integral Eq. and Op. Th. 13 (1990), 235–242.
J. Ueberberg: Zur Spektralinvariantz von Algebren von Pseudodifferen- tialoperatoren in der L P-Theorie, Manuscripta Math. 61 (1988), 459–475.
J. L. Walsh: Lnterpolation and approximation, Amer. Math. Soc. Coll. Publ. 20 (1969).
H. Widom: Singular integral equations in L P, Trans. Amer. Math. Soc. 97 (1960), 131–161.
N. Wiener, A. Wintner: Fourier Stieltjes transforms and singular infinite convolutions, Amer. J. of Math. 60 (1938), 513–522.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Birkhäuser Boston
About this chapter
Cite this chapter
Alvarez, J. (1999). Spectra of pseudo-differential operators in the Hörmander class. In: Bray, W.O., Stanojević, Č.V. (eds) Analysis of Divergence. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2236-1_13
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2236-1_13
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-7467-4
Online ISBN: 978-1-4612-2236-1
eBook Packages: Springer Book Archive