Hilbert transform and singular integrals on the spaces of tempered ultradistributions
Kamiński, Andrzej ; Perišić, Dušanka ; Pilipović, Stevan
Banach Center Publications, Tome 51 (2000), p. 139-153 / Harvested from The Polish Digital Mathematics Library
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The Hilbert transform on the spaces S'*(Rd) of tempered ultradistributions is defined, uniquely in the sense of hyperfunctions, as the composition of the classical Hilbert transform with the operators of multiplying and dividing a function by a certain elliptic ultrapolynomial. We show that the Hilbert transform of tempered ultradistributions defined in this way preserves important properties of the classical Hilbert transform. We also give definitions and prove properties of singular integral operators with odd and even kernels on the spaces S'*(Rd), whose special cases are the Hilbert transform and Riesz operators.

Publié le : 2000-01-01
EUDML-ID : urn:eudml:doc:209069 
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     author = {Kami\'nski, Andrzej and Peri\v si\'c, Du\v sanka and Pilipovi\'c, Stevan},
     title = {Hilbert transform and singular integrals on the spaces of tempered ultradistributions},
     journal = {Banach Center Publications},
     volume = {51},
     year = {2000},
     pages = {139-153},
     zbl = {0973.46032},
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Kamiński, Andrzej; Perišić, Dušanka; Pilipović, Stevan. Hilbert transform and singular integrals on the spaces of tempered ultradistributions. Banach Center Publications, Tome 51 (2000) pp. 139-153. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv53z1p139bwm/

[000] [1] E. J. Beltrami and M. R. Wohlers, Distributions and Boundary Values of Analytic Functions, Academic Press, New York, 1966. | Zbl 0186.19202

[001] [2] S. Ishikawa, Generalized Hilbert transforms in tempered distributions, Tokyo J. Math. 10 (1987), 119-132. | Zbl 0692.46040

[002] [3] A. Kamiński, D. Perišić and S. Pilipović, Integral transforms on the spaces of tempered ultradistributions, Demonstratio Math. 33 (2000), to appear. | Zbl 0973.46032

[003] [4] S. Koizumi, On the singular integrals I-VI, Proc. Japan Acad. 34 (1958), 193-198; 235-240; 594-598; 653-656; 35 (1959), 1-6; 323-328.

[004] [5] S. Koizumi, On the Hilbert transform I, II, J. Fac. Sci. Hokkaido Univ. Ser. I, 14 (1959), 153-224; 15 (1960), 93-130.

[005] [6] H. Komatsu, Ultradistributions, I, J. Fac. Sci. Univ. Tokyo Sect. IA 20 (1973), 25-105.

[006] [7] H. Komatsu, Ultradistributions, II, J. Fac. Sci. Univ. Tokyo Sect. IA 24 (1977), 607-628.

[007] [8] D. Kovačević and S. Pilipović, Structural properties of the spaces of tempered ultradistributions, in: Complex Analysis and Generalized Functions, Varna 1991, Publ. House of the Bugarian Academy of Sciences, Sofia 1993, 169-184. | Zbl 0795.46028

[008] [9] J. N. Pandey, An extension of the Gelfand-Shilov technique for Hilbert transforms, Journal of Applicable Analysis 13 (1982), 279-290. | Zbl 0468.46029

[009] [10] B. E. Petersen, Introduction to the Fourier Transform & Pseudo-differential Operators, Pitman, Boston, 1983. | Zbl 0523.35001

[010] [11] S. Pilipović, Hilbert transformation of Beurling ultradistributions, Rend. Sem. Mat. Univ. Padova 77 (1987), 1-13. | Zbl 0636.46043

[011] [12] S. Pilipović, Tempered ultradistributions, Boll. Un. Mat. Ital. (7) 2-B (1988), 235-251. | Zbl 0657.46030

[012] [13] S. Pilipović, Beurling-Gevrey tempered ultradistributions as boundary values, Portug. Math. 48 (1991), 483-504. | Zbl 0763.46030

[013] [14] S. Pilipović, Characterization of bounded sets in spaces of ultradistributions, Proc. Amer. Math. Soc. 120 (1994), 1191-1206. | Zbl 0816.46026

[014] [15] O. P. Singh and J. N. Pandey, The n-dimensional Hilbert transform of distributions, its inversion and applications, Can. J. Math 42 (1990), 239-258. | Zbl 0715.46016

[015] [16] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, 1971. | Zbl 0232.42007

[016] [17] H.-G. Tillmann, Randverteilungen analytischer Funktionen und Distributionen, Math. Zeitschr. 59 (1953), 61-83.

[017] [18] V. S. Vladimirov, Generalized Functions in Mathematical Physics, Mir, Moscow, 1979. | Zbl 0515.46033

[018] [19] J. Wloka, Grundräume und verallgemeinerte Funktionen, Lecture Notes in Math. 82, Springer, Berlin, 1969.

[019] [20] B. Ziemian, The modified Cauchy transformation with applications to generalized Taylor expansions, Studia Math. 102 (1992), 1-24. | Zbl 0815.46035