L p -boundedness for pseudodifferential operators with non-smooth symbols and applications

Gianluca Garello; Alessandro Morando

Bollettino dell'Unione Matematica Italiana (2005)

  • Volume: 8-B, Issue: 2, page 461-503
  • ISSN: 0392-4033

Abstract

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Starting from a general formulation of the characterization by dyadic crowns of Sobolev spaces, the authors give a result of L p continuity for pseudodifferential operators whose symbol a(x,ξ) is non smooth with respect to x and whose derivatives with respect to ξ have a decay of order ρ with 0 < ρ 1 . The algebra property for some classes of weighted Sobolev spaces is proved and an application to multi - quasi - elliptic semilinear equations is given.

How to cite

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Garello, Gianluca, and Morando, Alessandro. "$L^p$-boundedness for pseudodifferential operators with non-smooth symbols and applications." Bollettino dell'Unione Matematica Italiana 8-B.2 (2005): 461-503. <http://eudml.org/doc/195845>.

@article{Garello2005,
abstract = {Starting from a general formulation of the characterization by dyadic crowns of Sobolev spaces, the authors give a result of $L^\{p\}$ continuity for pseudodifferential operators whose symbol a(x,ξ) is non smooth with respect to x and whose derivatives with respect to ξ have a decay of order ρ with $0 < \rho \leq 1$. The algebra property for some classes of weighted Sobolev spaces is proved and an application to multi - quasi - elliptic semilinear equations is given.},
author = {Garello, Gianluca, Morando, Alessandro},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {461-503},
publisher = {Unione Matematica Italiana},
title = {$L^p$-boundedness for pseudodifferential operators with non-smooth symbols and applications},
url = {http://eudml.org/doc/195845},
volume = {8-B},
year = {2005},
}

TY - JOUR
AU - Garello, Gianluca
AU - Morando, Alessandro
TI - $L^p$-boundedness for pseudodifferential operators with non-smooth symbols and applications
JO - Bollettino dell'Unione Matematica Italiana
DA - 2005/6//
PB - Unione Matematica Italiana
VL - 8-B
IS - 2
SP - 461
EP - 503
AB - Starting from a general formulation of the characterization by dyadic crowns of Sobolev spaces, the authors give a result of $L^{p}$ continuity for pseudodifferential operators whose symbol a(x,ξ) is non smooth with respect to x and whose derivatives with respect to ξ have a decay of order ρ with $0 < \rho \leq 1$. The algebra property for some classes of weighted Sobolev spaces is proved and an application to multi - quasi - elliptic semilinear equations is given.
LA - eng
UR - http://eudml.org/doc/195845
ER -

References

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  1. BEALS, M. - REEDS, M. C., Microlocal regularity theorems for non smooth pseudodifferential operators and applications to non linear problems, Trans. Am. Math. Soc., 285 (1984), 159-184. Zbl0562.35093MR748836
  2. BOGGIATTO, P. - BUZANO, E. - RODINO, L., Global Hypoellipticity and Spectral Theory, Mathematical Research, Vol. 92, Akademie Verlag, Berlin, New York, 1996. Zbl0878.35001MR1435282
  3. BONY, J. M., Calcul simbolique et propagation des singularités pour les équations aux dérivées partielles non lineaires, Ann. Sc. Ec. Norm. Sup., 14 (1981), 161-205. Zbl0495.35024MR631751
  4. BONY, J. M. - CHEMIN, J. Y., Espaces fonctionnels associés au calcul de Weyl-Hörmander, Bull. Soc. Math. France, 122 (1994), 77-118. Zbl0798.35172MR1259109
  5. CALDERÓN, A. P., Intermediate spaces and interpolation, the complex method, Studia Math., 24 (1964), 113-190. Zbl0204.13703MR167830
  6. COIFMAN, R. - MEYER, Y., Au delà des opérateurs pseudo-differentiels, Astérisque57, Soc. Math. France, 1978. Zbl0483.35082MR518170
  7. EGOROV, Y. V. - SCHULZE, B. W., Pseudo-differential operators, singularities, applications, Operator Theory: Advances and Applications, 93, Birkhäuser Verlag, Basel, 1997. Zbl0877.35141MR1443430
  8. FEFFERMAN, C., L p bounds for pseudodifferential operators, Israel J. Math., 14 (1973), 413-417. Zbl0259.47045MR336453
  9. GARELLO, G., Generalized Sobolev algebras and regularity for solutions of multiquasi-elliptic semi linear equations, Comm. in Appl. Analysis, 3 (4) (1999), 563-574. Zbl0933.35204MR1706710
  10. GARELLO, G., Pseudodifferential operators with symbols in weighted Sobolev spaces and regularity for non linear partial differential equations, Math. Nachr., 239-240 (2001), 62-79. Zbl1027.35170MR1905664
  11. GARELLO, G. - MORANDO, A., L p -bounded pseudodifferential operators and regularity for multi-quasi-elliptic equations, to appear in Integr. equ. oper. theory. Zbl1082.35175
  12. GINDIKIN, S. - VOLEVICH, L. R., The method of Newton’s Polyhedron in the theory of partial differential equations, Coll. Mathematics and its Applications, Kluwer Academic Publishers, 1992. Zbl0779.35001MR1256484
  13. HELFFER, B., Théorie spectrale pour des opérateurs globalement elliptiques, Soc. Math. de France, Astérisque, 1984. Zbl0541.35002MR743094
  14. HÖRMANDER, L., The Weyl calculus of pseudodifferential operators, Comm. Pure Appl. Math., 32 (3) (1979), 359-443. Zbl0388.47032MR517939
  15. HÖRMANDER, L., The analysis of linear partial differential operators II. Differential operators with constant coefficients, Grundlehren der Mathematischen Wissenschaften, vol. 257, Springer-Verlag, Berlin, 1983. Zbl0521.35002MR705278
  16. LIZORKIN, P. I., L p , L q -multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR, 152 (1963), 808-811. Zbl0156.12902MR154057
  17. MARSCHALL, J., Pseudodifferential operators with non regular symbols of the class S ρ , δ m , Comm. in Part. Diff. Eq., 12 (8) (1987), 921-965. Zbl0621.47048MR891745
  18. MARSCHALL, J., Pseudo-differential operators with coefficients in Sobolev spaces, Trans. Amer. Math. Soc., 307 (1) (1988), 335-361. Zbl0679.35088
  19. SHUBIN, M. A., Pseudodifferential operators and spectral theory, Springer-Verlag, Berlin, 1987. Zbl0616.47040MR883081
  20. STEIN, E. M., Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J.1970. Zbl0207.13501MR290095
  21. TAYLOR, M. E., Pseudodifferential Operators, Princeton Univ. Press1981. Zbl0453.47026MR618463
  22. TAYLOR, M. E., Pseudodifferential operators and nonlinear PDE, Birkhäuser, Basel-Boston-Berlin, 1991. Zbl0746.35062MR1121019
  23. TRIEBEL, H., Interpolation theory, function spaces, differential operators, VEB, Berlin, 1977. Zbl0387.46033MR503903
  24. TRIEBEL, H., Theory of Function Spaces, Birkhäuser Verlag, Basel, Boston, Stuttgart, 1983. Zbl0763.46025MR781540
  25. TRIEBEL, H., General Function Spaces, I. Decomposition method, Math. Nachr., 79 (1977), 167-179. Zbl0374.46026MR628009
  26. TRIEBEL, H., General Function Spaces, II. Inequalities of Plancherel-Pólya- Nikol'skij type. L p -spaces of analytic functions; 0 < p , J. Approximation Theory, 19 (1977), 154-175. Zbl0344.46062MR628147
  27. TRIEBEL, H., General Function Spaces, III. Spaces B p , q g x and F p , q g x , 1 < p < : basic properties, Anal. Math., 3 (3) (1977), 221-249. Zbl0374.46027MR628468
  28. TRIEBEL, H., General Function Spaces, IV. Spaces B p , q g x and F p , q g x , 1 < p < : special properties, Anal. Math., 3 (4) (1977), 299-315. Zbl0374.46028MR628469
  29. TRIEBEL, H., General Function Spaces, V. The spaces B p , q g x and F p , q g x the case 0 < p < , Math. Nachr., 87 (1979), 129-152. Zbl0414.46025MR536420
  30. WONG, M. W., An introduction to pseudo-differential operators, 2nd ed., World Scientific Publishing Co., Inc., River Edge, NJ, 1999. Zbl0753.35134MR1698573

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