Some remarks on convolution equations

Berenstein, C. A. ; Dostal, M. A.

Annales de l'Institut Fourier, Tome 23 (1973), p. 55-73 / Harvested from Numdam

Résumé

Par voie d’une description de la topologie des espaces ${\bf E}^{\prime }(\Omega )$ ($\Omega $ ouvert convexe dans $R^n$) via la transformation de Fourier, c’est-à-dire leurs structures analytiques uniformes, on arrive à une formule qui décrit l’enveloppe convexe du support singulier d’une distribution $T$, $T\in {\bf E}^{\prime }$. On donne des applications à une classe des distributions qui satisfont à l’égalité

cv.sing.supp.S*T=cv.sing.supp.S+cv.sing.supp.T

pour toutes $S\in {\bf E}^{\prime }$.

Abstract

Using a description of the topology of the spaces ${\bf E}^{\prime }(\Omega )$ ($\Omega $ open convex subset of $R^n$) via the Fourier transform, namely their analytically uniform structures, we arrive at a formula describing the convex hull of the singular support of a distribution $T$, $T\in {\bf E}^{\prime }$. We give applications to a class of distributions $T$ satisfying

cv.sing.supp.S*T=cv.sing.supp.S+cv.sing.supp.T

for all $S\in {\bf E}^{\prime }$.

Détail

Publié le : 1973-01-01

MR 49 #5822 | Zbl 0241.46039

DOI : https://doi.org/10.5802/aif.444

Bibliographie

[1] A.D. Aleksandrov, “Die innere Geometrie der konvexen Flächen”, Berlin, 1955. | Zbl 0065.15102

[2] I.Ja. Bakel'Man, “Geometric methods of solutions of elliptic equations” (in Russian), Moscow, 1964.

[3] C.A. Berenstein and M.A. Dostal, Topological properties of the analytically uniform spaces, Trans. Amer. Math. Soc., 154 (1971), 493-513. | MR 53 #1252 | Zbl 0208.37403

[4] C.A. Berenstein and M.A. Dostal, Fourier transforms of the Beurling classes Dw, E'w', Bull. Amer. Math. Soc., 77 (1971), 963-967. | MR 44 #5769 | Zbl 0229.46038

[5] C.A. Berenstein and M.A. Dostal, “Analytically uniform spaces and their applications to convolution equations”, Lecture Notes in Math., vol. 256, Springer-Verlag, 1972. | MR 58 #12344 | Zbl 0237.47025

[6] N. Bourbaki, “Espaces vectoriels topologiques”, Eléments de mathématique, Livre V, Hermann et Cie, Paris, 1953, 1955.

[7] M.A. Dostal, On Fourier image of the singular support of a distribution, Czech. Math. J., 16 (1966), 231-237. | MR 33 #4593 | Zbl 0151.18203

[8] M.A. Dostal, An analogue of a theorem of Vladimir Bernstein and its applications to singular supports of distributions, Proc. London Math. Soc., 19 (1969), p. 553-576. | MR 40 #3302 | Zbl 0176.10302

[9] M.A. Dostal, A complex characterization of the Schwartz space D ($ATT$), Math. Ann., 195 (1972), p. 175-191. | MR 45 #888 | Zbl 0213.40002

[10] L. Ehrenpreis, Solutions of some problems of division I, Amer. J. Math. 76 (1954), 883-893. | MR 16,834a | Zbl 0056.10601

[11] L. Ehrenpreis, Solutions of some problems of division II, Amer. J. Math., 77 (1955), 286-292. | MR 16,1123a | Zbl 0064.11504

[12] L. Hörmander, On the range of convolution operators, Ann. Math., 76 (1962), 148-170. | MR 25 #5379 | Zbl 0109.08501

[13] L. Hörmander, Supports and singular supports of convolutions, Acta Math., 110 (1965), 279-302. | MR 27 #4070 | Zbl 0188.19406

[14] B. Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier, 6 (1955-1956), 271-355. | Numdam | MR 19,280a | Zbl 0071.09002

[15] B. Malgrange, Sur la propagation de la régularité des solutions des équations à coefficients constants, Bull. Math. Soc. Sci. Math. Phys. R.P. Roumaine, 3 (1959), 432-440. | Zbl 0109.32002

[16] K. Reidemeister, Uber die singulären Randpunkte eines konvexen Körpers, Math. Ann., 83 (1921), 116-118. | JFM 48.0835.03

[17] W. Rudin, “Function theory in polydisks”, W.A. Benjamin, Inc., New York, 1969. | Zbl 0177.34101

[18] L. Schwartz, “Théorie des distributions”, I, II, Hermann et Cie, Paris, 1950, 1953. | Zbl 0037.07301