This is an announcement for the paper "Lorentz spaces with variable exponents" by Henning Kempka and Jan Vybiral.

Abstract: We introduce Lorentz spaces $L_{p(\cdot),q}(\R^n)$ and $L_{p(\cdot),q(\cdot)}(\R^n)$ with variable exponents. We prove several basic properties of these spaces including embeddings and the identity $L_{p(\cdot),p(\cdot)}(\R^n)=L_{p(\cdot)}(\R^n)$. We also show that these spaces arise through real interpolation between $L_{\p}(\R^n)$ and $L_\infty(\R^n)$. Furthermore, we answer in a negative way the question posed in \cite{DHN} whether the Marcinkiewicz interpolation theorem holds in the frame of Lebesgue spaces with variable integrability.

Archive classification: math.FA

Submitted from: henning.kempka@uni-jena.de

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1210.1738

or

http://arXiv.org/abs/1210.1738