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.