On some properties of superreflexive Besov spaces

This paper contains results concerning superreflective Besov spaces $B^s_{p,q}(\mathbb{R}^n)$. Namely, expressions for convexity moduli and smoothness moduli with respect to the “canonical” norms are derived, and properties related to the finite representability of Banach spaces and linear compact operators in $B^s_{p,q}(\mathbb{R}^n)$ are examined. Additionally, inequalities of the Prus–Smarzewski type for arbitrary equivalent norms and inequalities of the James–Gurariy type are presented. Based on the latter, two-sided estimates for the norms of elements in $B^s_{p,q}(\mathbb{R}^n)$ can be obtained in terms of the expansion coefficients of these elements in unconditional normalized Schauder bases.