Fundamental solutions for hypoelliptic differential operators depending analytically on a parameter
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- by Frank Mantlik
- Trans. Amer. Math. Soc. 334 (1992), 245-257
- DOI: https://doi.org/10.1090/S0002-9947-1992-1107027-5
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Abstract:
Let $P(\lambda ,D) = \sum \nolimits _{|\alpha | \leq m} {{a_\alpha }(\lambda ){D^\alpha }}$ be a differential operator with constant coefficients ${a_\alpha }$ depending analytically on a parameter $\lambda$. Assume that each $P(\lambda ,D)$ is hypoelliptic and that the strength of $P(\lambda ,D)$ is independent of $\lambda$. Under this condition we show that there exists a regular fundamental solution of $P(\lambda ,D)$ which also depends analytically on $\lambda$.References
- Hans Grauert, On Levi’s problem and the imbedding of real-analytic manifolds, Ann. of Math. (2) 68 (1958), 460–472. MR 98847, DOI 10.2307/1970257
- Lars Hörmander, On the division of distributions by polynomials, Ark. Mat. 3 (1958), 555–568. MR 124734, DOI 10.1007/BF02589517
- Lars Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0203075
- Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035, DOI 10.1007/978-3-642-96750-4
- Hans Jarchow, Locally convex spaces, Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart, 1981. MR 632257
- Jürgen Leiterer, Banach coherent analytic Fréchet sheaves, Math. Nachr. 85 (1978), 91–109. MR 517643, DOI 10.1002/mana.19780850108
- S. Łojasiewicz, Sur le problème de la division, Studia Math. 18 (1959), 87–136 (French). MR 107168, DOI 10.4064/sm-18-1-87-136
- I. I. Priwalow, Randeigenschaften analytischer Funktionen, Hochschulbücher für Mathematik, Band 25, VEB Deutscher Verlag der Wissenschaften, Berlin, 1956 (German). Zweite, unter Redaktion von A. I. Markuschewitsch überarbeitete und ergänzte Auflage. MR 0083565
- François Trèves, Opérateurs différentiels hypoelliptiques, Ann. Inst. Fourier (Grenoble) 9 (1959), 1–73 (French). MR 114056 —, Un théorème sur les équations aux dérivées partielles à coefficients constants dépendant de paramètres, Bull. Soc. Math. France 90 (1962), 473-486.
- François Trèves, Fundamental solutions of linear partial differential equations with constant coefficients depending on parameters, Amer. J. Math. 84 (1962), 561–577. MR 149084, DOI 10.2307/2372862
Bibliographic Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 334 (1992), 245-257
- MSC: Primary 35H05; Secondary 35B30
- DOI: https://doi.org/10.1090/S0002-9947-1992-1107027-5
- MathSciNet review: 1107027