Remarks on a Subspace of Morrey Spaces

Abstract

Let $p$, $\lambda$ be real numbers such that $1<p<\infty$, and $0<\lambda<1$. Also let $L^{p,\lambda}(\mathbb{T})$ be Morrey spaces on the unit circle $\mathbb{T}$, and $L^{p,\lambda}_0(\mathbb{T})$ the closure of $C(\mathbb{T})$ in $L^{p,\lambda}(\mathbb{T})$. Zorko [7] gave the predual $Z^{q,\lambda}(\mathbb{T})\ (1/p+1/q=1)$ of $L^{p,\lambda}(\mathbb{T})$. In this article, we show a property of $L^{p,\lambda}_0(\mathbb{T})$ and prove in detail that $L_0^{p,\lambda}(\mathbb{T})$ is the predual of $Z^{q,\lambda}(\mathbb{T})$, whose fact is stated in Adams-Xiao [1].