Lorentz spaces with variable exponents

We introduce Lorentz spaces $L_{p(\cdot ),q}({\mathbb{R}}^n)$ and $L_{p(\cdot ),q(\cdot )}({\mathbb{R}}^n)$ with variable exponents. We prove several basic properties of these spaces including embeddings and the identity $L_{p(\cdot ),p(\cdot )}({\mathbb{R}}^n)=L_{p(\cdot )}({\mathbb{R}}^n)$. We also show that these spaces arise through real interpolation between $L_{\text{p}}({\mathbb{R}}^n)$ and $L_{\mathrm{\infty }}({\mathbb{R}}^n)$. 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.