Asymptotic Expansions of Finite Hankel Transforms and the Surjectivity of Convolution Operators

Yasunori Okada ; Hideshi Yamane

Symmetry, integrability and geometry: methods and applications, Tome 20 (2024)

Résumé

A compactly supported distribution is called invertible in the sense of Ehrenpreis–{Hörmander} if the convolution with it induces a surjection from $\mathcal{C}^{\infty}(\mathbb{R}^{n})$ to itself. We give sufficient conditions for radial functions to be invertible. Our analysis is based on the asymptotic expansions of finite Hankel transforms. The dominant term may be the contribution from the origin or from the boundary of the support of the function. For the proof, we propose a new method to calculate the asymptotic expansions of finite Hankel transforms of functions with singularities at a point other than the origin.

Détail

Keywords: asymptotic expansion, Hankel transform, invertibility.
Mots-clés : convolution

Bibliographie

[1] Abramczuk W., “A class of surjective convolution operators”, Pacific J. Math., 110 (1984), 1–7 | DOI | MR

[2] Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia Math. Appl., 71, Cambridge University Press, Cambridge, 1999 | DOI | MR | Zbl

[3] Berenstein C.A., Dostal M.A., “On convolution equations. I”, L'Analyse Harmonique dans le Domaine Complexe, Actes Table Ronde Internat. (Centre Nat. Recherche Sci., Montpellier), Lecture Notes in Math., 336, Springer, Berlin, 1972, 79–94 | DOI | MR

[4] Christensen J., Gonzalez F., Kakehi T., “Surjectivity of mean value operators on noncompact symmetric spaces”, J. Funct. Anal., 272 (2017), 3610–3646, arXiv: 1608.01869 | DOI | MR | Zbl

[5] Debrouwere A., Kalmes T., “Linear topological invariants for kernels of convolution and differential operators”, J. Funct. Anal., 284 (2023), 109886, 20 pp., arXiv: 2204.11733 | DOI | MR | Zbl

[6] Ehrenpreis L., “Solution of some problems of division. I Division by a polynomial of derivation”, Amer. J. Math., 76 (1954), 883–903 | DOI | MR | Zbl

[7] Ehrenpreis L., “Solutions of some problems of division. {III} {D}ivision in the spaces, ${\mathcal D}'$, ${\mathcal H}$, ${\mathcal Q}_A$, ${\mathcal O}$”, Amer. J. Math., 78 (1956), 685–715 | DOI | MR | Zbl

[8] Ehrenpreis L., “Solution of some problems of division. IV Invertible and elliptic operators”, Amer. J. Math., 82 (1960), 522–588 | DOI | MR | Zbl

[9] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Tables of integral transforms, v. II, McGraw-Hill Book Co., Inc., New York, 1954 | MR

[10] Gradshteyn I.S., Ryzhik I.M., Table of integrals, series, and products, 8th ed., Academic Press, Amsterdam, 2015 | MR | Zbl

[11] Hörmander L., “On the range of convolution operators”, Ann. of Math., 76 (1962), 148–170 | DOI | MR | Zbl

[12] Hörmander L., The analysis of linear partial differential operators, v. II, Grundlehren Math. Wiss., 257, Differential operators with constant coefficients, Springer, Berlin, 1983 | DOI | MR | Zbl

[13] Lim K.-T., The spherical mean value operators on Euclidean and hyperbolic spaces, Ph.D. Thesis, Tufts University, Medford, MA, 2012 https://dl.tufts.edu/concern/pdfs/pz50h7447

[14] Malgrange B., “Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution”, Ann. Inst. Fourier (Grenoble), 6 (1955/56), 271–355 | DOI | MR

[15] Musin I.Kh., “Perturbation of a surjective convolution operator”, Ufa Math. J., 8 (2016), 123–130 | DOI | MR | Zbl

[16] Okada Y., Yamane H., “Generalized spherical mean value operators on Euclidean space”, Tsukuba J. Math., 45 (2021), 37–50, arXiv: 2003.10005 | DOI | MR | Zbl

[17] Olver F.W.J., Olde Daalhuis A.B., Lozier D.W., Schneider B.I., Boisvert R.F., Clark C.W., Miller B.R., Saunders B.V., Cohl H.S., McClain M.A. (Editors), NIST Digital Library of Mathematical Functions, Release 1.2.0 of 2024-03-15, https://dlmf.nist.gov

[18] Stein E.M., Weiss G., Introduction to Fourier analysis on Euclidean spaces, Princeton Math. Ser., 32, Princeton University Press, Princeton, NJ, 1971 | DOI | MR | Zbl

[19] Watson G.N., A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, 1944 | MR | Zbl

[20] Wong R., “Error bounds for asymptotic expansions of Hankel transforms”, SIAM J. Math. Anal., 7 (1976), 799–808 | DOI | MR | Zbl

[21] Wong R., “Asymptotic expansions of Hankel transforms of functions with logarithmic singularities”, Comput. Math. Appl., 3 (1977), 271–286 | DOI | MR | Zbl

[22] Wong R., “Error bounds for asymptotic expansions of integrals”, SIAM Rev., 22 (1980), 401–435 | DOI | MR | Zbl