Spaces defined by the level function and their duals

Gord Sinnamon

Studia Mathematica (1994)

  • Volume: 111, Issue: 1, page 19-52
  • ISSN: 0039-3223

Abstract

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The classical level function construction of Halperin and Lorentz is extended to Lebesgue spaces with general measures. The construction is also carried farther. In particular, the level function is considered as a monotone map on its natural domain, a superspace of L p . These domains are shown to be Banach spaces which, although closely tied to L p spaces, are not reflexive. A related construction is given which characterizes their dual spaces.

How to cite

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Sinnamon, Gord. "Spaces defined by the level function and their duals." Studia Mathematica 111.1 (1994): 19-52. <http://eudml.org/doc/216117>.

@article{Sinnamon1994,
abstract = {The classical level function construction of Halperin and Lorentz is extended to Lebesgue spaces with general measures. The construction is also carried farther. In particular, the level function is considered as a monotone map on its natural domain, a superspace of $L^p$. These domains are shown to be Banach spaces which, although closely tied to $L^p$ spaces, are not reflexive. A related construction is given which characterizes their dual spaces.},
author = {Sinnamon, Gord},
journal = {Studia Mathematica},
keywords = {function spaces; Hölder's inequality; Hardy's inequality; dual spaces; Hölder inequality; level function construction of Halperin and Lorentz; Lebesgue spaces with general measures; monotone map; superspace of },
language = {eng},
number = {1},
pages = {19-52},
title = {Spaces defined by the level function and their duals},
url = {http://eudml.org/doc/216117},
volume = {111},
year = {1994},
}

TY - JOUR
AU - Sinnamon, Gord
TI - Spaces defined by the level function and their duals
JO - Studia Mathematica
PY - 1994
VL - 111
IS - 1
SP - 19
EP - 52
AB - The classical level function construction of Halperin and Lorentz is extended to Lebesgue spaces with general measures. The construction is also carried farther. In particular, the level function is considered as a monotone map on its natural domain, a superspace of $L^p$. These domains are shown to be Banach spaces which, although closely tied to $L^p$ spaces, are not reflexive. A related construction is given which characterizes their dual spaces.
LA - eng
KW - function spaces; Hölder's inequality; Hardy's inequality; dual spaces; Hölder inequality; level function construction of Halperin and Lorentz; Lebesgue spaces with general measures; monotone map; superspace of
UR - http://eudml.org/doc/216117
ER -

References

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  1. [1] G. Bennett, Some elementary inequalities, III, Quart. J. Math. Oxford Ser. (2) 42 (1991), 149-174. Zbl0751.26007
  2. [2] J. S. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), 405-408. Zbl0402.26006
  3. [3] I. Halperin, Function spaces, Canad. J. Math. 5 (1953), 273-288. Zbl0052.11303
  4. [4] G. G. Lorentz, Bernstein Polynomials, Univ. of Toronto Press, Toronto, 1953. 
  5. [5] V. G. Maz'ja, Sobolev Spaces, Springer, Berlin, 1985. 
  6. [6] B. Muckenhoupt, Hardy's inequality with weights, Studia Math. 44 (1972), 31-38. Zbl0236.26015
  7. [7] H. L. Royden, Real Analysis, 2nd ed., Macmillan, New York, 1968. Zbl0197.03501
  8. [8] G. J. Sinnamon, Operators on Lebesgue spaces with general measures, Doctoral Thesis, McMaster Univ., 1987. 
  9. [9] G. J. Sinnamon, Weighted Hardy and Opial-type inequalities, J. Math. Anal. Appl. 160 (1991), 434-445. Zbl0756.26011
  10. [10] G. J. Sinnamon, Interpolation of spaces defined by the level function, in: Harmonic Analysis, ICM-90 Satellite Proceedings, Springer, Tokyo, 1991, 190-193. Zbl0783.46018

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