On Some Properties of Superreflexive Besov Spaces

Abstract

This paper contains results concerning superreflective Besov spaces $$B_{{p,q}}^{s}({{\mathbb{R}}^{n}})$$ . 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 $$B_{{p,q}}^{s}({{\mathbb{R}}^{n}})$$ 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 $$B_{{p,q}}^{s}({{\mathbb{R}}^{n}})$$ can be obtained in terms of the expansion coefficients of these elements in unconditional normalized Schauder bases.