Generalized Hankel conjugate transformations on rearrangement invariant spaces

Kerman, R. A.

doi:10.1090/S0002-9947-1977-0450882-4

Generalized Hankel conjugate transformations on rearrangement invariant spaces
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by R. A. Kerman

Trans. Amer. Math. Soc. 232 (1977), 111-130

DOI: https://doi.org/10.1090/S0002-9947-1977-0450882-4

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Abstract:

The boundedness properties of the generalized Hankel conjugate transformations ${H_\lambda }$ on certain weighted Lebesgue spaces are studied. These are used to establish a boundedness criterion for the ${H_\lambda }$ on the more general class of rearrangement invariant spaces. The positive operators in terms of which the criterion is given are used to construct pairs of spaces between which the ${H_\lambda }$ are continuous; in particular, a natural analogue of a well-known result of Zygmund concerning the classical conjugate function operator is obtained for the ${H_\lambda }$.

References

Kenneth F. Andersen, Norm inequalities for ultraspherical and Hankel conjugate functions, Canadian J. Math. 27 (1975), 162–171. MR 358203, DOI 10.4153/CJM-1975-021-7
Colin Bennett, A Hausdorff-Young theorem for rearrangement-invariant spaces, Pacific J. Math. 47 (1973), 311–328. MR 338654, DOI 10.2140/pjm.1973.47.311
D. W. Boyd, The Hilbert transform on rearrangement-invariant spaces, Canadian J. Math. 19 (1967), 599–616. MR 212512, DOI 10.4153/CJM-1967-053-7
David W. Boyd, Indices of function spaces and their relationship to interpolation, Canadian J. Math. 21 (1969), 1245–1254. MR 412788, DOI 10.4153/CJM-1969-137-x
A.-P. Calderón, Spaces between $L^{1}$ and $L^{\infty }$ and the theorem of Marcinkiewicz, Studia Math. 26 (1966), 273–299. MR 203444, DOI 10.4064/sm-26-3-301-304
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR 0065685

Boundedness criteria for generalized Hankel conjugate transformations

B. Muckenhoupt and E. M. Stein, Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118 (1965), 17–92. MR 199636, DOI 10.1090/S0002-9947-1965-0199636-9
R. O’Neil and G. Weiss, The Hilbert transform and rearrangement of functions, Studia Math. 23 (1963), 189–198. MR 160084, DOI 10.4064/sm-23-2-189-198
P. G. Rooney, On the ranges of certain fractional integrals, Canadian J. Math. 24 (1972), 1198–1216. MR 310147, DOI 10.4153/CJM-1972-130-9
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469, DOI 10.1017/CBO9780511608759
A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587