This is an announcement for the paper "New formulas for decreasing rearrangements and a class of spaces" by Anna Kaminska and Yves Raynaud.
Abstract: Using a nonlinear version of the well known Hardy-Littlewood inequalities, we derive new formulas for decreasing rearrangements of functions and sequences in the context of convex functions. We use these formulas for deducing several properties of the modular functionals defining the function and sequence spaces $M_{\varphi,w}$ and $m_{\varphi,w}$ respectively, introduced earlier in \cite{HKM} for describing the K"othe dual of ordinary Orlicz-Lorentz spaces in a large variety of cases ($\varphi$ is an Orlicz function and $w$ a {\it decreasing} weight). We study these $M_{\varphi,w}$ classes in the most general setting, where they may even not be linear, and identify their K"othe duals with ordinary (Banach) Orlicz-Lorentz spaces. We introduce a new class of rearrangement invariant Banach spaces $\mathcal{M}_{\varphi,w}$ which proves to be the K"othe biduals of the $M_{\varphi,w}$ classes. In the case when the class $M_{\varphi,w}$ is a separable quasi-Banach space, $\mathcal{M}_{\varphi,w}$ is its Banach envelope.
Archive classification: math.FA
Mathematics Subject Classification: 26D07, 39B62, 42B25, 46B10, 46E30
Remarks: 25 pages
Submitted from: kaminska@memphis.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1301.4465
or
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alspach＠math.okstate.edu