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Dual spaces of Lorentz-Zygmund type in optimal interpolation

Dual spaces of Lorentz-Zygmund type in optimal interpolation Abstract. We find the dual norms for spaces of Lorentz-Zygmund type and for other spaces from [P1] involved in optimal weak type interpolation. This allows us to prove optimal interpolation theorems for the left endpoint of the interpolation segment and to extend all results to spaces with underlying interval ð0; yÞ. 1991 Mathematics Subject Classification: 46B70, 46E30; 46E35, 46M35. 1 Introduction The classical Marcinkiewicz interpolation theorem and its generalizations (known as theorems of weak type interpolation, see [B], [KPS] etc.) require from spaces under consideration to be ``suciently distant'' from the endpoint spaces. Analytically, such conditions have a form of certain strict inequalities for the corresponding Boyd indices. Recall that these indices are defined for an arbitrary rearrangement-invariant space E by pE ¼ lim t!0 ln dE ðtÞ ; ln t qE ¼ lim t!y ln dE ðtÞ ; ln t dE ðtÞ ¼ sup f AE k f ðs=tÞkE : k f ðsÞkE In the present paper we consider another kind of weak type interpolation, where the spaces are just in extremal positions, with the Boyd indices inadmissible for classical theorems. More exactly, we consider the (quasi)linear operators T, which are of two weak types ða; bÞ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Dual spaces of Lorentz-Zygmund type in optimal interpolation

Forum Mathematicum , Volume 16 (3) – Mar 30, 2004

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References (8)

Publisher
de Gruyter
Copyright
Copyright © 2004 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2004.017
Publisher site
See Article on Publisher Site

Abstract

Abstract. We find the dual norms for spaces of Lorentz-Zygmund type and for other spaces from [P1] involved in optimal weak type interpolation. This allows us to prove optimal interpolation theorems for the left endpoint of the interpolation segment and to extend all results to spaces with underlying interval ð0; yÞ. 1991 Mathematics Subject Classification: 46B70, 46E30; 46E35, 46M35. 1 Introduction The classical Marcinkiewicz interpolation theorem and its generalizations (known as theorems of weak type interpolation, see [B], [KPS] etc.) require from spaces under consideration to be ``suciently distant'' from the endpoint spaces. Analytically, such conditions have a form of certain strict inequalities for the corresponding Boyd indices. Recall that these indices are defined for an arbitrary rearrangement-invariant space E by pE ¼ lim t!0 ln dE ðtÞ ; ln t qE ¼ lim t!y ln dE ðtÞ ; ln t dE ðtÞ ¼ sup f AE k f ðs=tÞkE : k f ðsÞkE In the present paper we consider another kind of weak type interpolation, where the spaces are just in extremal positions, with the Boyd indices inadmissible for classical theorems. More exactly, we consider the (quasi)linear operators T, which are of two weak types ða; bÞ

Journal

Forum Mathematicumde Gruyter

Published: Mar 30, 2004

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