Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Generalized Hankel conjugate transformations on rearrangement invariant spaces


Author: R. A. Kerman
Journal: Trans. Amer. Math. Soc. 232 (1977), 111-130
MSC: Primary 42A40; Secondary 44A25
DOI: https://doi.org/10.1090/S0002-9947-1977-0450882-4
MathSciNet review: 0450882
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] K. F. Andersen, Norm inequalities for ultraspherical and Hankel conjugate functions, Canad. J. Math. 27 (1975), 162-171. MR 50 # 10668. MR 0358203 (50:10668)
  • [2] C. Bennett, A Hausdorff-Young theorem for rearrangement-invariant spaces, Pacific J. Math. 47 (1973), 311-328. MR 49 #3418. MR 0338654 (49:3418)
  • [3] D. W. Boyd, The Hilbert transform on rearrangement-invariant spaces, Canad. J. Math. 19 (1967), 599-616. MR 35 #3383. MR 0212512 (35:3383)
  • [4] -, Indices of function spaces and their relationship to interpolation, Canad. J. Math. 21 (1969), 1245-1254. MR 0412788 (54:909)
  • [5] A.-P. Calderón, Spaces between $ {L^1}$ and $ {L^\infty }$ and the theorem of Marcinkiewicz, Studia Math. 26 (1966), 273-299. MR 34 #3295. MR 0203444 (34:3295)
  • [6] A. Erdélyi et al., Tables of integral transforms, Vol. II, McGraw-Hill, New York, 1954. MR 16, 468. MR 0065685 (16:468c)
  • [7] R. A. Kerman, Boundedness criteria for generalized Hankel conjugate transformations (manuscript).
  • [8] B. Muckenhoupt and E. M. Stein, Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118 (1965), 17-92. MR 33 #7779. MR 0199636 (33:7779)
  • [9] R. O'Neil and G. Weiss, The Hilbert transform and rearrangement of functions, Studia Math. 23 (1963), 189-198. MR 28 #3298. MR 0160084 (28:3298)
  • [10] P. G. Rooney, On the ranges of certain fractional integrals, Canad. J. Math. 24 (1972), 1198-1216. MR 46 #9249. MR 0310147 (46:9249)
  • [11] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Univ. Press, Cambridge; Macmillan, New York, 1944. MR 6, 64. MR 0010746 (6:64a)
  • [12] E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed., Cambridge Univ. Press, New York, 1927; reprint, 1962. MR 31 #2375. MR 1424469 (97k:01072)
  • [13] A. Zygmund, Trigonometric series, Vols. I, II, Monografie Mat., Bd. 5, PWN, Warsaw, 1935; 2nd ed., revised and reprinted, Cambridge Univ. Press, New York, 1968. MR 38 #4882. MR 0236587 (38:4882)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 42A40, 44A25

Retrieve articles in all journals with MSC: 42A40, 44A25


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0450882-4
Keywords: Hankel conjugate transformation, boundedness, rearrangement invariant space, Lebesgue space, Lorentz space, Orlicz space
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society