Geodesic
On an integral equation of Šub-Sizonenko
Glasgow mathematical journal, Tome 24 (1983) no. 2, pp. 207-210

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The integral equation of the title isIt 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 byIn 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 formulaWe 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. 
Rooney, P. G. On an integral equation of Šub-Sizonenko. Glasgow mathematical journal, Tome 24 (1983) no. 2, pp. 207-210. doi: 10.1017/S0017089500005309
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[1] 1.Erdélyi, A. et al. , Tables of integral transforms I, (McGraw-Hill, 1954). Google Scholar

[2] 2.Rooney, P. G., On the ranges of certain fractional integrals, Canad. J. Math. 24 (1952), 1198–1216. Google Scholar | DOI

[3] 3.Rooney, P. G., Multipliers for the Mellin transformation, Canad. Math. Bull, (to appear). Google Scholar

[4] 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.) Google Scholar | DOI

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