Hostname: page-component-cb9f654ff-65tv2 Total loading time: 0 Render date: 2025-08-19T17:03:42.679Z Has data issue: false hasContentIssue false
  • English
  • Français

On the range and inversion of fractional integrals in weighted spaces

Published online by Cambridge University Press:  14 November 2011

Kenneth F. Andersen
Kenneth F. Andersen
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada
Get access Rights & Permissions [Opens in a new window]

Synopsis

The weight functions w(x) for which the Riemann fractional integral operator Iα is bounded from the Lebesgue space Lp(wp) into Lq(wq), l/q = l/p −, have been characterized by Muckenhoupt and Wheeden. In this paper, we prove an inversion formula for Iα in the context of these weighted spaces and we also characterize the range of Iα as a subset of Lq(wq) Similar results are proved for other fractional integrals. These results may be viewed as weighted analogues of certain results of Stein and Zygmund, Herson and Heywood, Heywood, and Kober who considered the unweighted case, w(x) = l.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 92 , Issue 1-2 , 1982 , pp. 51 - 64
Copyright
Copyright © Royal Society of Edinburgh 1982

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.)

Article purchase

Temporarily unavailable

References

1Coifman, R. R. and Fefferman, C.. Weighted norm inequalities for maximal functions and singular integrals. Studia Math. 51 (1974), 241–250.Google Scholar
2Dunford, N. and Schwartz, J. T.. Linear Operators, Part I (New York: Interscience, 1958).Google Scholar
3Hardy, G. H. and Littlewood, J. E.. Some properties of fractional integrals I. Math. Zeit. 27 (1928), 565–606.CrossRefGoogle Scholar
4Herson, D. L. J. and Heywood, P.. On the range of some fractional integrals. J. London Math. Soc. 8 (1974), 607–614.Google Scholar
5Heywood, P.. On a modification of the Hilbert transform. J. London Math. Soc. 42 (1967), 641–645.CrossRefGoogle Scholar
6Heywood, P.. On the inversion of fractional integrals. J. London Math. Soc. 3 (1971), 531–538.Google Scholar
7Hunt, R., Muckenhoupt, B. and Wheeden, R. L.. Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Amer. Math. Soc. 176 (1973), 227–251.Google Scholar
8Kober, H.. New properties of the Weyl extended integral. Proc. London Math. Soc. 21 (1970), 557–575.CrossRefGoogle Scholar
9Muckenhoupt, B.. Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165 (1972), 207–226.Google Scholar
10Muckenhoupt, B. and Wheeden, R. L.. Weighted norm inequalities for fractional integrals. Trans. Amer. Math. Soc. 192 (1974), 261–274.Google Scholar
11Okikiolu, G. O.. A generalisation of the Hilbert transform. J. London Math. Soc. 40 (1965), 27–30.CrossRefGoogle Scholar
12Rudin, W.. Real and Complex Analysis (New York: McGraw Hill, 1966).Google Scholar
13Stein, E. M.. Singular Integrals and the Differentiability Properties of Functions (Princeton Univ. Press, 1970).Google Scholar
14Stein, E. M. and Zygmund, A.. On the fractional differentiability of functions. Proc. London Math. Soc. 14A (1965), 249–264.CrossRefGoogle Scholar
15Welland, G. V.. Weighted norm inequalities for fractional integrals. Proc. Amer. Math. Soc. 51 (1975), 143–148.CrossRefGoogle Scholar
16Zygmund, A.. Trigonometric Series, Vol. I and II combined (Cambridge Univ. Press, 1968).Google Scholar