Inequalities of Gagliardo-Nirenberg type and estimates for the moduli of continuity

In this paper a study is made of multiplicative inequalities of Gagliardo-Nirenberg type that connect partial moduli of continuity and partial derivatives of functions with respect to a fixed variable in different Lorentz norms. The main results are expressed by estimates and the exponents pi and si satisfy certain conditions. In particular, these estimates imply optimal inequalities involving Besov norms and Lorentz norms. The limit case p1=s1=1 and estimates in terms of total variation are also studied