Abstract
This paper presents a general definition of pseudo-differential operators of type 1, 1; the definition is shown to be the largest one that is both compatible with negligible operators and stable under vanishing frequency modulation. Elaborating counter-examples of Ching, Hörmander and Parenti-Rodino, type 1, 1-operators with unclosable graphs are proved to exist; others are shown to lack the microlocal property as they flip the wavefront set of an almost nowhere differentiable function. In contrast the definition is shown to imply the pseudo-local property, so type 1, 1-operators cannot create singularities but only change their nature. The familiar rule that the support of the argument is transported by the support of the distribution kernel is generalised to arbitrary type 1, 1-operators. A similar spectral support rule is also proved. As no restrictions appear for classical type 1, 0-operators, this is a new result which in many cases makes it unnecessary to reduce to elementary symbols. As an important tool, a convergent sequence of distributions is said to converge regularly if it moreover converges as smooth functions outside the singular support of the limit. This notion is shown to allow limit processes in extended versions of the formula relating operators and kernels.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J.-M. Bony, Calcul symbolique et propagations des singularités pour les équations aux dérivées partielles non linéaires, Ann. scient. Éc. Norm. Sup. 14 (1981), 209–246.
G. Bourdaud, Sur les opérateurs pseudo-différentiels à coefficients peu reguliers, Thèse, Univ. de Paris-Sud, 1983.
-, Une algèbre maximale d’opérateurs pseudo-différentiels, Comm. Partial Differential Equations 13 (1988), no. 9, 1059–1083.
C.H. Ching, Pseudo-differential operators with nonregular symbols, J. Differential Equations 11 (1972), 436–447.
R.R. Coifman and Y. Meyer, Au delà des opérateurs pseudo-différentiels, Astérisque, vol. 57, Société Mathématique de France, Paris, 1978.
M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777–799.
M. Frazier and B. Jawerth, A discrete transform and decomposition of distribution spaces, J. Funct. Anal. 93 (1990), 34–170.
G. Garello, Microlocal properties for pseudodifferential operators of type 1, 1, Comm. Partial Differential Equations 19 (1994), 791–801.
L. Hörmander, The Analysis of Linear Partial Differential Operators, Grundlehren der mathematischen Wissenschaften, Spring er-Ver lag, Berlin, 1983, 1985.
-, Pseudo-differential operators of type 1, 1, Comm. Partial Differential Equations 13 (1988), no. 9, 1085–1111.
-, Continuity of pseudo-differential operators of type 1,1, Comm. Partial Differential Equations 14 (1989), no. 2, 231–243.
-, Lectures on Nonlinear Differential Equations, Mathématiques & applications, vol. 26, Springer-Verlag, Berlin, 1997.
J. Johnsen, Pointwise multiplication of Besov and Triebel-Lizorkin spaces, Math. Nachr. 175 (1995), 85–133.
-, Domains of type 1, 1 operators: a case for Triebel-Lizorkin spaces, C. R. Acad. Sci. Paris Sér. I Math. 339 (2004), no. 2, 115–118.
-, Domains of pseudo-differential operators: a case for the Trieb el-Liz orkin spaces, J. Function Spaces Appl. 3 (2005), 263–286.
-, Simple proof s of nowhere-differentiability for Weierstrass’ function and cases of slow growth, Preprint R-2008-02, Aalborg University, 2008.
Y. Meyer, Régularité des solutions des équations aux dérivées partielles non linéaires (d’après J.-M. Bony), Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math. 842, Springer, Berlin, 1981, 293–302.
-, Remarques sur un théorème de J.-M. Bony, in Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980), 1981, 1–20.
C. Parenti and L. Rodino, A pseudo differential operator which shifts the wave front set, Proc. Amer. Math. Soc. 72 (1978), 251–257.
T. Runst, Pseudodifferential operators of the“exotic” class L1 0,1 in spaces of Besov and Trieb el-Lizorkin type, Ann. Global Anal. Geom. 3 (1985), no. 1, 13–28.
L. Schwartz, Théorie des Distributions, Revised and Enlarged ed., Hermann, Paris, 1966.
M.A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, 1987, Translated from the 1978 Russian original by Stig I. Andersson.
X. Saint Raymond, Elementary Introduction to the Theory of Pseudodifferential Operators, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1991.
E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993.
R.H. Torres, Continuity properties of pseudodifferential operators of type 1, 1, Comm. Partial Differential Equations 15 (1990), 1313–1328.
H. Triebel, Theory of Function Spaces, Monographs in mathematics, 78, Birkhäuser Verlag, Basel, 1983.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Birkhäuser Verlag Basel/Switzerland
About this chapter
Cite this chapter
Johnsen, J. (2008). Type 1,1-Operators Defined by Vanishing Frequency Modulation. In: Rodino, L., Wong, M.W. (eds) New Developments in Pseudo-Differential Operators. Operator Theory: Advances and Applications, vol 189. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8969-7_10
Download citation
DOI: https://doi.org/10.1007/978-3-7643-8969-7_10
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-8968-0
Online ISBN: 978-3-7643-8969-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)