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 of the form
$$ \biggl(\int_\delta^\infty[h^{-\theta r}\omega_j^r(f;h)_{p,s}]^s\,\frac{dh}h\biggr)^{1/s}\le c\|f\|_{p_0,s_0}^{1-\theta}[\delta^{-r}\omega_j^r(f;\delta)_{p_1,s_1}]^\theta, $$
where $0<\theta<1$,
$$ \frac1p=\frac{1-\theta}{p_0}+\frac{\theta}{p_1}\,, \qquad \frac1s=\frac{1-\theta}{s_0}+\frac{\theta}{s_1}\,, $$
and the exponents $p_i$ and $s_i$ satisfy certain conditions. In particular, these estimates imply optimal inequalities involving Besov norms and Lorentz norms. The limit case $p_1=s_1=1$ and estimates in terms of total variation are also studied.