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

TL;DR

Let $1\le p<\infty$ and $0<\lambda<1$. The paper shows that on the unit circle $\mathbb{T}$ the Morrey space $L^{p,\lambda}(\mathbb{T})$ cannot be renormed to be rearrangement-invariant by exhibiting equimeasurable $f,g:\mathbb{T}\to\mathbb{R}$ with $g\in L^{p,\lambda}(\mathbb{T})$ but $f\notin L^{p,\lambda}(\mathbb{T})$. The proof provides an explicit construction of $f$ and $g$ and verifies the divergent Morrey norm for $f$ while bounding it for $g$ through arc-based estimates over relevant arcs. Consequently, the result gives a negative answer to rearrangement-invariant renormings for Morrey spaces on $\mathbb{T}$, aligning with related observations on $\mathbb{R}$ and clarifying the behavior of these spaces on compact groups.

Abstract

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

Theorems & Definitions (1)

• Theorem 1.1