Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-06T11:57:34.031Z Has data issue: false hasContentIssue false

The finite Hilbert transform in weighted spaces

Published online by Cambridge University Press:  14 November 2011

Kari Astala
Affiliation:
Department of Mathematics, University of Helsinki, PL 4 (Hallituskatu 15), 00014 University of Helsinki, Finland
Lassi Päivärinta
Affiliation:
Department of Mathematical Sciences, University of Oulu, Linnanmaa, 90570 Oulu, Finland
Eero Saksman
Affiliation:
Department of Mathematics, University of Helsinki, PL 4 (Hallituskatu 15), 00014 University of Helsinki, Finland

Abstract

The mapping properties of the finite Hilbert-transform (respectively the Hilbert transform on the half axis) are studied. Invertibility, surjectivity, injectivity and bounded ness from below of the transform are characterised in general weighted spaces. The results are applied to the restriction of the operator with logarithmic kernel.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Carlemann, T.. Sur la résolution de certaines équations intégrates. Ark. Mat., Astro, och Fysik 16 (1992).Google Scholar
2Garcia-Cuerva, J. and Francia, J. Rubio de. Weighted Norm Inequalities and Related Topics (Amsterdam: North-Holland, 1985).Google Scholar
3Garnett, J. B.. Bounded Analytic Functions (San Diego: Academic Press, 1981).Google Scholar
4Gradshteyn, I. S. and Ryzhik, I. M.. Table of integrals, series, and products (New York: Academic Press, 1980).Google Scholar
5Hsiao, G. and MacCamy, R. C.. Solutions of boundary value problems by integral equations of the first kind. SIAM Review 15 (1973), 687704.CrossRefGoogle Scholar
6Hunt, R. A., Muchenhoupt, B. and Wheeden, R. L.. Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Amer. Math. Soc. 176 (1973), 227–52.CrossRefGoogle Scholar
7Koppelman, W. and Pincus, J. D.. Spectral representations for finite Hilbert transformations. Math. Z. 71 (1959), 399407.CrossRefGoogle Scholar
8Okada, S. and Elliot, D.. The finite Hilbert transform in ℒ2. Math. Nachr. 153 (1991), 5768.CrossRefGoogle Scholar
9Rooney, P. G.. On the spectrum of an internal operator. Glasgow Math. J. 28 (1986), 59.CrossRefGoogle Scholar
10Tricomi, F. G.. On the finite Hilbert transform. Quart. J. Math. 2 (1951), 199211.CrossRefGoogle Scholar
11Tricomi, F. G.. Integral Equations (New York: Interscience, 1957).Google Scholar