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Journal of Function Spaces and Applications
Volume 4 (2006), Issue 3, Pages 275-304

New classes of rearrangement-invariant spaces appearing in extreme cases of weak interpolation

1Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel
2Departamento de Matemática Aplicada, Facultad de Informática, Universidad de Murcia, Campus de Espinardo, 30100 Espinardo (Murcia), Spain

Received 1 December 2005

Academic Editor: Fernando Cobos

Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study weak type interpolation for ultrasymmetric spaces L?,E i.e., having the norm ??(t)f*(t)?E˜, where ?(t) is any quasiconcave function and E˜ is arbitrary rearrangement-invariant space with respect to the measure dt/t. When spaces L?,E are not “too close” to the endpoint spaces of interpolation (in the sense of Boyd), the optimal interpolation theorem was stated in [13]. The case of “too close” spaces was studied in [15] with results which are optimal, but only among ultrasymmetric spaces. In this paper we find better interpolation results, involving new types of rearrangement-invariant spaces, A?,b,E and B?,b,E, which are described and investigated in detail.