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On an integral equation of Šub-Sizonenko

Published online by Cambridge University Press:  18 May 2009

P. G. Rooney
P. G. Rooney
Affiliation:
University of Toronto, Toronto, Canada, M5S 1A1.
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The integral equation of the title is

It was studied in [4], though h(x) was written as x-1g(x-1) there, and using a method involving orthogonal Watson transformations, it was shown there that if h ∈ L2(0, ∞), then the equation has a solution f ∈ L2(0, ∞), and that / is given by

In this paper, using the techniques of [3], we shall show that the equation can be solved for ℎ in the space ℒμ, p of [3] for 1 ≤ p < ∘, μ > 0, and that for these spaces, which include L2(0, ∘), f is given by the simpler formula

We shall further show that these results can be extended to the spaces ℒw, μ, p of [3]. This forms the content of our theorem below.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 24 , Issue 2 , July 1983 , pp. 207 - 210
Copyright
Copyright © Glasgow Mathematical Journal Trust 1983

References

REFERENCES

1.Erdélyi, A. et al. , Tables of integral transforms I, (McGraw-Hill, 1954).Google Scholar
2.Rooney, P. G., On the ranges of certain fractional integrals, Canad. J. Math. 24 (1952), 1198–1216.CrossRefGoogle Scholar
3.Rooney, P. G., Multipliers for the Mellin transformation, Canad. Math. Bull, (to appear).Google Scholar
4.Šub-Sizonenko, J. A., Inversion of an integral operator by the method of expansion with respect to orthogonal Watson operators, Siberian Math. J. 20 (1979), 318–321. (Also Sibirsk. Mat. Ž. 20 (1979), 445–448.)CrossRefGoogle Scholar