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Pseudodifferential operators with coefficients in Sobolev spaces


Author: Jürgen Marschall
Journal: Trans. Amer. Math. Soc. 307 (1988), 335-361
MSC: Primary 35S05; Secondary 35A27, 47G05
DOI: https://doi.org/10.1090/S0002-9947-1988-0936820-3
MathSciNet review: 936820
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Abstract: Pseudo-differential operators with coefficients in Sobolev spaces $ {H^{r,q}},1 \leqslant q \leqslant \infty $, and their adjoints are studied on Hardy-Sobolev spaces $ {H^{s,p}},\;0 < p \leqslant \infty $. A symbolic calculus for these operators is developed, and the microlocal properties are studied. Finally, the invariance under coordinate transformations is proved.


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  • [1] M. Beals, Propagation of smoothness for nonlinear second-order strictly hyperbolic differential equations, Proc. Sympos. Pure Math., vol. 43, Amer. Math. Soc., Providence, R.I., 1985, pp. 21-44. MR 812282 (87b:35107)
  • [2] M. Beals and M. Reed, Microlocal regularity theorems for non-smooth pseudo-differential operators and applications to non-linear problems, Trans. Amer. Math. Soc. 285 (1984), 159-184. MR 748836 (86a:35156)
  • [3] J.-M. Bony, Calcul symbolique et propagation des singularités pour les equations aux dérivées partielles nonlinéaries, Ann. Sci. École Norm. Sup. 14 (1981), 209-246. MR 631751 (84h:35177)
  • [4] G. Bourdaud, $ {L^p}$-estimates for certain nonregular pseudo-differential operators, Comm. Partial Differential Equations 7 (1982), 1023-1033. MR 673825 (83m:47039)
  • [5] R. R. Coifman and Y. Meyer, Au delà des opérateurs pseudo-différentiels, Astérisque 57 (1978), 1-185. MR 518170 (81b:47061)
  • [6] J. Franke, On the spaces $ F_{p,q}^s$ of Triebel-Lizorkin type: pointwise multipliers and spaces on domains, Math. Nachr. 125 (1986), 29-68. MR 847350 (88e:46029)
  • [7] D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979), 27-42. MR 523600 (80h:46052)
  • [8] L. Hörmander, Pseudodifferential operators and hypoelliptic equations, Proc. Sympos. Pure Math., vol. 10, Amer. Math. Soc., Providence, R.I., 1966, pp. 138-183.
  • [9] B. Jawerth, Some observations on Besov and Triebel-Lizorkin spaces, Math. Scand. 40 (1977), 94-104. MR 0454618 (56:12867)
  • [10] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426. MR 0131498 (24:A1348)
  • [11] H. Kumano Go and M. Nagase, Pseudo-differential operators with nonregular symbols and applications, Funkcial. Ekvac. 21 (1978), 151-192. MR 518297 (80b:47066)
  • [12] J. Marschall, Pseudo-differential operators with nonregular symbols, Thesis, Freie Universität Berlin, 1985.
  • [13] -, Some remarks on Triebel spaces, Studia Math. 87 (1987), 79-92. MR 924763 (89a:46076)
  • [14] -, Pseudo-differential operators with nonregular symbols of the class $ S_{\rho ,\delta }^m$, Comm. Partial Differential Equations 12 (1987), 921-965.
  • [15] Y. Meyer, Remarques sur un Théorème de J. M. Bony, Rend. Circ. Mat. Palermo 2 (1981), Suppl. 1, 1-20. MR 639462 (83b:35169)
  • [16] J. Peetre, New thoughts on Besov spaces, Duke Univ. Math. Series 1, Durham, 1976. MR 0461123 (57:1108)
  • [17] J. Rauch, Singularities of solutions to semilinear wave equations, J. Math. Pures Appl. 58 (1979), 299-308. MR 544255 (83c:35078)
  • [18] Th. Runst, Para-differential operators in spaces of Triebel-Lizorkin and Besov type, Z. Anal. Anwendungen 4 (1985), 557-573. MR 818984 (87c:35038)
  • [19] R. S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967), 1031-1060. MR 0215084 (35:5927)
  • [20] F. Treves, Introduction to pseudo-differential and Fourier integral operators, vol. 1, Plenum, New York, 1980.
  • [21] H. Triebel, Theory of function spaces, Birkhäuser Verlag, Boston, Mass., 1983. MR 781540 (86j:46026)
  • [22] M. Yamazaki, A quasi-homogeneous version of paradifferential operators, I. Boundedness on spaces of Besov type, J. Fac. Sci. Univ. Tokyo Sect. IA 33 (1986), 131-174. II. A symbol calculus, ibid. 311-345.
  • [23] -, A quasi-homogeneous version of the microlocal analysis for nonlinear partial differential equations, Preprint.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0936820-3
Keywords: Pseudo-differential operators, nonregular symbols, Sobolev spaces, microlocal analysis
Article copyright: © Copyright 1988 American Mathematical Society

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