Morrey spaces over the unit circle cannot be renormed to become rearrangement-invariant

Abstract

Let $1\le p<\infty$ and $0<\lambda<1$. We consider the classical Morrey space $L^{p,\lambda}(\mathbb{T})$ over the unit circle $\mathbb{T}$. We show that there are equimeasurable functions $f,g:\mathbb{T}\to\mathbb{R}$ such that $g\in L^{p,\lambda}(\mathbb{T})$ but $f\notin L^{p,\lambda}(\mathbb{T})$. This implies that the the space $L^{p,\lambda}(\mathbb{T})$ cannot be renormed to become rearrangement-invariant.