Abstract. The purpose of this note is to study the relationship between the Besov spaces $B_p$ (or $B_p^\#$) and the recently introduced classes of functions $Q_{q,0}$ (or $Q_{q,0}^\#$). We establish necessary and sufficient conditions for a lacunary series to belong to $B_p$ and use this result to show that our generalizations of two basic theorems involving Besov spaces are best possible. There are some unexpected differences between the results for analytic Besov spaces, $B_p$, and those for spherical Besov spaces, $B_p^\#$; these differences arise when $1< p< 2$. As a by-product of these results we also obtain a new characterization of the family of little normal functions.

AMS Subject Classification (1991): 30H05, 30B10, 30D45

Received March 16, 1994 and in revised form October 19, 1994. (Registered under 5617/2009.)