Hypoelliptic convolution equations in 𝐾’_{𝑝}, 𝑝>1

Sampson, G.; Zieleźny, Z.

doi:10.1090/S0002-9947-1976-0425607-8

Hypoelliptic convolution equations in ,

Authors: G. Sampson and Z. Zieleźny
Journal: Trans. Amer. Math. Soc. 223 (1976), 133-154
MSC: Primary 46F10; Secondary 47G05
DOI: https://doi.org/10.1090/S0002-9947-1976-0425607-8
MathSciNet review: 0425607
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider convolution equations in the space ${K'_p},p > 1$ , of distributions which ``grow'' no faster than $\exp (k\vert x{\vert^p})$ for some constant k.

Our main result is a complete characterization of hypoelliptic convolution operators in ${K'_p}$ in terms of their Fourier transforms.

[1] L. Ehrenpreis, Solution of some problems of division. IV. Invertible and elliptic operators, Amer. J. Math. 82 (1960), 522–588. MR 119082, https://doi.org/10.2307/2372971
[2] G. I. Èskin, A generalization of a theorem of Paley-Wiener-Schwarz, Uspehi Mat. Nauk 16 (1961), no. 1 (97), 185–188 (Russian). MR 0131124
[3] Morisuke Hasumi, Note on the 𝑛-dimensional tempered ultra-distributions, Tohoku Math. J. (2) 13 (1961), 94–104. MR 131759, https://doi.org/10.2748/tmj/1178244354
[4] Lars Hörmander, Hypoelliptic convolution equations, Math. Scand. 9 (1961), 178–184. MR 139838, https://doi.org/10.7146/math.scand.a-10633
[5] L. Schwartz, Séminaire de Schwartz 1953/54. Produits tensoriels topologiques d'espaces vectoriels topologiques. Espaces vectoriels topologiques nucléaires, Secrétariat mathématique, Paris, 1954. MR 17, 764.
[6] -, Théorie des distributions. Tomes I, II, Actualités Sci. Indust., nos. 1091, 1122, Hermann, Paris, 1950, 1951. MR 12, 31; 833.
[7] Z. Zieleźny, Hypoelliptic and entire elliptic convolution equations in subspaces of the space of distributions. I, Studia Math. 28 (1966/67), 317–332. MR 222476, https://doi.org/10.4064/sm-28-3-317-332
[8] Z. Zieleźny, Hypoelliptic and entire elliptic convolution equations in subspaces of the space of distributions. II, Studia Math. 32 (1969), 47–59. MR 248521, https://doi.org/10.4064/sm-32-1-47-59