Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-12-02T23:25:14.996Z Has data issue: false hasContentIssue false
  • English
  • Français

Norm Inequalities for Ultraspherical and Hankel Conjugate Functions

Published online by Cambridge University Press:  20 November 2018

Kenneth F. Andersen
Kenneth F. Andersen*
Affiliation:
University of Alberta, Edmonton, Alberta
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The notion of conjugate functions associated with ultraspherical expansions and their continuous analogues, the Hankel transforms, was introduced by Muckenhoupt and Stein [14], to which we refer the reader for general background and an excellent discussion of the motivation underlying these notions. The operation of passing from a given function to its conjugate is in many ways analogous to the passage from a function to its Hilbert transform, indeed, Muckenhoupt and Stein proved, among other things, that these operations acting on appropriate weighted Lebesgue spaces, Lp(𝝁), satisfy inequalities of M. Riesz type analogous to those satisfied by the Hilbert transform on the usual Lebesgue spaces, Lv( — ∞, ∞).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 1 , 01 February 1975 , pp. 162 - 171
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Boyd, D. W., The Hilbert transform on rearrangement-invariant spaces. Can. J. Math. 19 (1967), 599–616.Google Scholar
2. Boyd, D. W., Indices of function spaces and their relationship to interpolation, Can. J. Math. 21 (1969), 1245–1154.Google Scholar
3. Indices for the Orlicz spaces, Pacific J. Math. 38 (1971), 315–323.Google Scholar
4. Calderôn, A. P., Spaces between L1 and L °° and the theorem of Marcinkiewicz, Studia Math. 26 (1966), 273–299.Google Scholar
5. Halperin, I., Uniform convexity in function spaces, Duke Math. J. 21 (1954), 195–204.Google Scholar
6. Halperin, I., Reflexivity in the Injunction spaces, Duke Math. J. 21 (1954), 205–208.Google Scholar
7. Hunt, R. A., On L﹛p, q) spaces, Enseignement Math. 12 (1966), 249–276.Google Scholar
8. Kerman, R. A., Conjugate junctions and rearrangement invariant spaces, Ph.D. Thesis, University of Toronto, 1969. (see Dissertation Abstracts International 31 (1971), p. 6752-B).Google Scholar
9. Krasnosel'skii, M. A. and Rutickii, Ya. B., Convex functions and Orlicz spaces (N∞rdhoff, Groningen, 1961).Google Scholar
10. Lorentz, G. G., On the theory of spaces A, Pacific J. Math. 1 (1950), 411–429.Google Scholar
11. Lorentz, G. G., Some new function spaces, Ann. of Math. 51 (1950), 37–55.Google Scholar
12. Luxemburg, W. A. J., Banach function spaces, Ph.D. Thesis, Delft Technical University, Assen, 1956.Google Scholar
13. Luxemburg, W. A. J., Rearrangement invariant Banach function spaces, Queen's Papers in Pure and Applied Math. 10 (1967), 83–144.Google Scholar
14. Muckenhoupt, B. and Stein, E. M., Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118 (1965), 17–92.Google Scholar
15. Ryan, R., Conjugate functions in Orlicz spaces, Pacific J. Math. IS (1963), 1371–1377.Google Scholar
16. Zygmund, A., Trigonometrical Series, Vol. I, II (Cambridge University Press, Cambridge, 1968).Google Scholar