Ulyanov-type inequalities between Lorentz-Zygmund spaces

Abstract

We establish inequalities of Ulyanov-type for moduli of smoothness relating the source Lorentz-Zygmund space $ \, L^{p,r}(\log L)^{\alpha -\gamma},\, \gamma >0,$ and the target space $ \, L^{p^*,s}(\log L)^\alpha $ over $ \, {\mathbb R}^n$ if $ \, 1<p<p^*<\infty$ and over $ \, \mathbb{T}^n$ if $ \, 1<p \le p^*<\infty.$ The stronger logarithmic integrability (corresponding to $ \, L^{p^*,s}(\log L)^\alpha $ ) is balanced by an additional logarithmic smoothness.