Abstract
Distributions having compact support are represented as the boundary value of Cauchy and Poisson integrals corresponding to tubular radial domains TC in ℂn where C is an open convex cone. The Cauchy integral of U ε ɛ′ is shown to be an analytic function in TC which satifies a certain boundedness condition. All analytic functions in TC having this boundedness condition have a distributional boundary value which can be used to determine an ɛ′ distribution. The results are extended to vector valued distributions.
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References
BOCHNER, S., and W.T. MARTIN: Several complex variables. Princeton,N.J.: Princeton U. Press 1948.
BREMERMANN, H.J.: Schwartz distributions as boundary values of n-harmonic functions. J. D'Analyse Math. 14, 5–13 (1965).
BREMERMANN, H.J.: Distributions, complex variables, and Fourier transforms. Reading, Mass.: Addison-Wesley 1965.
BREMERMANN, H.J., and L. DURAND, III: On analytic continuation, multiplication, and Fourier transformations of Schwartz distributions. J. Math. Physics 2, 240–258 (1961).
CARMICHAEL, R.D.: Distributional boundary values of functions analytic in tubular radial domains. Indiana U. Math. J. (formerly J. Math. Mech.) 20, 843–853 (1971).
CARMICHAEL, R.D.: Generalized Cauchy and Poisson integrals and distributional boundary values. SIAM J. Math. Analysis (to appear).
GEL'FAND, I.M., and G.E. SHILOV: Generalized functions, vol. 1. New York: Academic Press 1964.
GROTHENDIECK, A.: Sur les espaces (F) et (DF). Summa Brasiliensis Math. 3, 57–123 (1954).
KORÁNYI, A.: A Poisson integral for homogeneous wedge domains. J. D'Analyse Math. 14, 275–284 (1965).
KÖTHE, G.: Die Randverteilungen analytischer Funktionen. Math. Z. 57, 13–33 (1952).
MITROVIĆ, D.: Plemelj formulas and analytic representation of distributions. Glasnik Mat. 3 (23), 231–239 (1968).
MITROVIĆ, D.: The Plemelj distributional formulas. Bull. Amer. Math. Soc. 77, 562–563 (1971).
SCHWARTZ, L.: Théorie des distributions. Paris: Hermann 1966.
SCHWARTZ, L.: Théorie des distributions a valeurs vectorielles. I. Ann. Inst. Fourier 7, 1–149 (1957).
SCHWARTZ, L.: Théorie des distributions a valeurs vectorielles. II. Ann. Inst. Fourier 8, 1–209 (1958).
SCHWARTZ, L.: Mathematics for the physical sciences. Reading, Mass.: Addison-Wesley 1966.
SIMMONS, G.F.: Introduction to topology and modern analysis. New York: McGraw-Hill 1963.
STEIN, E.M., and G. WEISS: Introduction to Fourier analysis on Euclidean spaces. Princeton, N.J.: Princeton U. Press 1971.
TILLMANN, H.G.: Randverteilungen analytischer Funktionen und Distributionen. Math. Z. 59, 61–83 (1953).
TILLMANN, H.G.: Distributionen als Randverteilungen analytischer Funktionen. II. Math. Z. 76, 5–21 (1961).
TILLMANN, H.G.: Darstellung der Schwartzschen Distributionen durch analytische Funktionen. Math. Z. 77, 106–124 (1961).
TILLMANN, H.G.: Darstellung vektorwertiger Distributionen durch holomorphe Funktionen. Math. Ann. 151, 286–295 (1963).
VLADIMIROV, V.S.: On functions holomorphic in tubular cones. Izv. Akad. Nauk SSSR Ser. Matem. 27, 75–100 (1963). (in Russian)
VLADIMIROV, V.S.: On the construction of envelopes of holomorphy for regions of a special type and their applications. Trudy Mat. Inst. Steklov 60, 101–144 (1961). (Translated in Amer. Math. Soc. Translation 48 (series 2), 107–150 (1966).)
VLADIMIROV, V. S.: Methods of the theory of functions of several complex variables. Cambridge, Mass.: M.I.T. Press 1966.
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Carmichael, R.D., Walker, W.W. Representation of distributions with compact support. Manuscripta Math 11, 305–338 (1974). https://doi.org/10.1007/BF01170235
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DOI: https://doi.org/10.1007/BF01170235