Lorentz spaces with variable exponents

Abstract

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 $L_{\p}(\R^n)$ and $L_{\infty }({\mathbb{R}}^n)$. Furthermore, we answer in a negative way the question posed in Diening, Hästö, and Nekvinda (2004) whether the Marcinkiewicz interpolation theorem holds in the frame of Lebesgue spaces with variable integrability.

References in corpus (1)

• Fractional, Maximal and Singular Operators in Variable Exponent Lorentz Spaces

Cited by in corpus (1)

• Complex interpolation of couple (X, BMO) for $A_1$-regular lattices