# Spaces defined by the level function and their duals

Studia Mathematica (1994)

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

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topSinnamon, 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

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