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.