\(L_p\)-boundedness of general index transforms. (English) Zbl 1091.44002
Lith. Math. J. 45, No. 1, 102-112 (2005) and Liet. Mat. Rink. 45, No. 1, 127-147 (2005).
Author’s summary: We establish the boundedness properties in \(L_p\) for a class of integral transformations with respect to an index of hypergeometric functions. In particular, by using the Riesz-Thorin interpolation theorem, we get the corresponding results in \(L_p (\mathbb R_+)\), \(1\leq p\leq 2\), for the Kontorovich-Lebedev, Mehler-Fock, and Olevskii index transforms. An inversion theorem is proved for a general index transformation. The case \(p=2\) is known as the Plancherel-type theory for this class of transformations.
Reviewer: H.-J. Glaeske (Jena)
MSC:
| 44A15 | Special integral transforms (Legendre, Hilbert, etc.) |
| 33C10 | Bessel and Airy functions, cylinder functions, \({}_0F_1\) |
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
Keywords:
Kontorovich-Lebedev transform; Hausdorff-Young inequality; Fourier transform; Mellin transform; Mehler-Fock transform; Olevskii transform; Plancherel theory; Mcdonald functions; Legendre functions; index transformationReferences:
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