Lorentz spaces with variable exponents
  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 Diening, H\"ast\"o, and Nekvinda (2004) whether the Marcinkiewicz interpolation theorem holds in the frame of Lebesgue spaces with variable integrability. 
Henning Kempka, Jan Vyb\'iral
math.FA