Extension and lacunas of solutions of linear partial differential equations
Annales de l'Institut Fourier, Volume 46 (1996) no. 2, pp. 429-464.

Let KQ be compact, convex sets in n with K and let P(D) be a linear, constant coefficient PDO. It is characterized in various ways when each zero solution of P(D) in the space (K) of all C -functions on K extends to a zero solution in (Q) resp. in ( n ). The most relevant characterizations are in terms of Phragmén-Lindelöf conditions on the zero variety of P in n and in terms of fundamental solutions for P(D) with lacunas.

Soient KQ des ensembles compacts, convexes dans n tel que K et soit P(D) un opérateur linéaire aux dérivées partielles à coefficients constants. On donne plusieurs conditions qui sont équivalentes au fait que chaque zéro-solution de P(D) dans l’espace (K) des fonctions C sur K au sens de Whitney a une extension comme zéro-solution dans (Q) ou dans ( n ). Des caractérisations intéressantes sont une condition du type de Phragmén-Lindelöf sur la variété de P dans n et une condition pour des solutions élémentaires pour P(D) avec lacunes.

@article{AIF_1996__46_2_429_0,
     author = {Franken, Uwe and Meise, Reinhold},
     title = {Extension and lacunas of solutions of linear partial differential equations},
     journal = {Annales de l'Institut Fourier},
     pages = {429--464},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {46},
     number = {2},
     year = {1996},
     doi = {10.5802/aif.1520},
     zbl = {0853.35022},
     mrnumber = {97h:35005},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1520/}
}
TY  - JOUR
AU  - Franken, Uwe
AU  - Meise, Reinhold
TI  - Extension and lacunas of solutions of linear partial differential equations
JO  - Annales de l'Institut Fourier
PY  - 1996
SP  - 429
EP  - 464
VL  - 46
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1520/
DO  - 10.5802/aif.1520
LA  - en
ID  - AIF_1996__46_2_429_0
ER  - 
%0 Journal Article
%A Franken, Uwe
%A Meise, Reinhold
%T Extension and lacunas of solutions of linear partial differential equations
%J Annales de l'Institut Fourier
%D 1996
%P 429-464
%V 46
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1520/
%R 10.5802/aif.1520
%G en
%F AIF_1996__46_2_429_0
Franken, Uwe; Meise, Reinhold. Extension and lacunas of solutions of linear partial differential equations. Annales de l'Institut Fourier, Volume 46 (1996) no. 2, pp. 429-464. doi : 10.5802/aif.1520. https://aif.centre-mersenne.org/articles/10.5802/aif.1520/

[1] C. Boiti and M. Nacinovich, Evolution and hyperbolic pairs, preprint.

[2] J.-M. Bony and M. Schapira, Existence et prolongement des solutions holomorphes des équations aux dérivées partielles, Inventiones Math., 17 (1972), 95-105. | MR | Zbl

[3] M.L. De Christoforis, Soluzioni con lacune di certi operatori differenziali lineari, Rendiconti Accademia Nazionale delle Scienze detta dei XL, Memorie di Matematica 102, vol. VIII (1984), 137-142.

[4] U. Franken, On the equivalence of holomorphic and plurisubharmonic Phragmén-Lindelöf principles, Michigan Math. J., 42 (1995), 163-173. | MR | Zbl

[5] U. Franken and R. Meise, Continuous linear right inverses for homogeneous linear partial differential operators on bounded convex open sets and extension of zero-solutions, Proceedings of the Trier work shop on “Functional Analysis”, S. Dierolf, S. Dineen, and P. Domanski (Eds.) de Gruyter (1996), to appear. | MR | Zbl

[6] S. Hansen, On the fundamental principle of L. Ehrenpreis, Banach Center Publ., 10 (1983), 185-201. | MR | Zbl

[7] L. Hörmander, On the existence of real analytic solutions of partial differential equations with constant coefficients, Invent. Math., 21 (1973), 151-183. | MR | Zbl

[8] L. Hörmander, The Analysis of Linear Partial Differential Operators I and II, Springer 1983. | Zbl

[9] A. Kaneko, Hartogs type extension theorem of real analytic solutions of linear partial differential equations with constant coefficients, Advances in the theory of Fréchet Spaces (T. Terzioglu, e.d.) NATO Adv. Sci. Inst., Ser. C : Math. Phys. Sci., 289 (1989), 63-72. | MR | Zbl

[10] C.O. Kiselman, Prolongement des solutions d'une équation aux dérivées partielles à coefficients constants, Bull. Soc. Math. France, 97 (1969), 329-356. | EuDML | Numdam | MR | Zbl

[11] M. Langenbruch, Extension of ultradifferentiable functions, Manuscripta Math., 83 (1994), 123-143. | EuDML | MR | Zbl

[12] O. Liess, Extension of zero solutions of linear partial differential operators, Darmstadt 1983, preprint.

[13] R. Meise and B.A. Taylor, Whitney's extension theorem for ultradifferentiable functions of Beurling type, Ark. Mat., 26 (1988), 265-287. | MR | Zbl

[14] R. Meise and B.A. Taylor, Linear extension operators for ultradifferentiable functions of Beurling type on compact sets, Amer. J. Math., 111 (1989), 309-337. | MR | Zbl

[15] R. Meise, B.A. Taylor and D. Vogt, Characterization of linear partial differential operators with constant coefficients that admit a continuous linear right inverse, Ann. Inst. Fourier, Grenoble, 40-3 (1990), 619-655. | EuDML | Numdam | MR | Zbl

[16] R. Meise, B.A. Taylor and D. Vogt, Equivalence of analytic and plurisubharmonic Phragmén-Lindelöf principles on algebraic varieties, Proceedings of Symposia in Pure Mathematics, 52 (1991), 287-308. | MR | Zbl

[17] R. Meise, B.A. Taylor and D. Vogt, Continuous linear right inverses for partial differential operators with constant coefficients and Phragmén-Lindelöf conditions, in “Functional Analysis”, K. D. Bierstedt, A. Pietsch, W. M. Ruess, and D. Vogt (Eds.) Lecture Notes in Pure and Applied Math., Vol. 150 Marcel Dekker, (1994), pp. 357-389. | MR | Zbl

[18] R. Meise, B.A. Taylor and D. Vogt, Phragmén-Lindelöf principles on algebraic varieties, J. Amer. Math. Soc., to appear. | MR | Zbl

[19] R. Meise, B.A. Taylor and D. Vogt, Continuous linear right inverses for partial differential operators of order 2 and fundamental solutions in half spaces, preprint. | EuDML | MR | Zbl

[20] R. Meise, D. Vogt, Einführung in die Funktionalanalysis, Vieweg, 1992. | MR | Zbl

[21] V. P. Palamodov, Linear Differential Operators with constant Coefficients, Springer, 1970. | MR | Zbl

[22] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. | MR | Zbl

[23] R. Schneider, Convex Bodies : the Minkowski Theory, Cambridge University Press, 1993. | MR | Zbl

[24] M. Tidten, Fortsetzung von C∞-Funktionen, welche auf einer abgeschlossenen Menge in ℝn definiert sind, Manuscripta Math., 27 (1979), 291-312. | EuDML | MR | Zbl

[25] H. Whitney, Analytic extension of differentiable Functions, defined on closed sets, Trans. Am. Math. Soc., 36 (1934), 63-89. | JFM | MR | Zbl

Cited by Sources: