On the relationship between block spaces and Orlicz spaces

Abstract

Let $1<q\le \infty$ and $v>-1$. Let $(X,d,μ)$ be an $s$-Ahlfors-regular quasi-metric measure space. Suppose that $B^{0,v}_q(X)$ is the block space which consists of all functions that admit a decomposition into $q$-blocks supported on balls. In this paper, we study the relationship between the block space $B^{0,v}_q(X)$ and the Orlicz-type space $L(\log^+\!\!L)^{1+v}(X)$. More precisely, we show that the block space $B_q^{0,v}(X)$ is a proper subspace of the Orlicz space $L(\log^+\!\!L)^{1+v}(X)$ for any fixed $1<q\le \infty$ and $v>-1$. Namely, $$B_q^{0,v}(X)\subsetneq L(\log^+\!\!L)^{1+v}(X),$$ which gives a confirmed answer to a longstanding open problem concerning the relationship between block spaces and Orlicz-type spaces on the unit sphere $\mathbb S^{n-1}$. We further show that $L(\log^+\!\!L)^{1+v}(X)$ is the smallest Orlicz-type space containing $B^{0,v}_{q}(X)$. We also introduce a generalized block space $\mathscr B_q^{0,v}(X)$ that depends only on the measure structure and show that this space is equivalent to the Orlicz space $L(\log^+\!\!L)^{1+v}(X)$ when $μ(X)<\infty$. Finally, we consider two special cases that further clarify the roles of the parameter $q$ and the logarithmic weight.