A characterization of Besov classes in terms of a new modulus of continuity (Q1707152)

It is well known that the classical Besov spaces \(B^\alpha_{p,\theta} (\mathbb R^n)\) with \(0<\alpha<1\), \(1\leq p \leq \infty\), \(1\leq \theta \leq \infty\), can be characterized in tems of the modulus of continuity \[ \omega_p (f,r) = \sup_{| h| \leq r} \| f(\cdot-h) - f(\cdot) \mid L_p (\mathbb R^n) \|. \] The author introduces some new moduli of continuity incorporating first derivatives (for example as the divergence of related vector-functions). It is the main aim of the paper to show that the above Besov spaces can be characterized in terms of these new moduli of continuity and that some new properties can be proved, including modifications of the Gagliardo-Nirenberg inequalities.