On Block Spaces on the Unit Sphere

In this paper, we establish several properties of the block spaces $B_q^{0,v}$. Our primary contribution resolves an open question regarding the relationship between $B_q^{0,v}$ and the Zygmund-type space $L(\log L{)}^{1+v}$ by demonstrating that $B_q^{0,v}$ is a proper subspace of $L(\log L{)}^{1+v}$. Furthermore, we prove that the union of all block spaces $B_q^{0,v}$ for $1<q\le \mathrm{\infty }$ is strictly contained within $L(\log L{)}^{1+v}$.