Hardy-Littlewood and Ulyanov inequalities

We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness $\omega_\alpha(f,t)_q$ and $\omega_\beta(f,t)_p$ for $0<p<q\le \infty$. A similar problem for the generalized $K$-functionals and their realizations between the couples $(L_p, W_p^\psi)$ and $(L_q, W_q^\varphi)$ is also solved. The main tool is the new Hardy-Littlewood-Nikol'skii inequalities. More precisely, we obtained the asymptotic behavior of the quantity $$\sup_{T_n} \frac{\Vert \mathcal{D}(\psi)(T_n)\Vert_q}{\Vert \mathcal{D}(\varphi)(T_n)\Vert_p},\qquad 0<p<q\le \infty,$$ where the supremum is taken over all nontrivial trigonometric polynomials $T_n$ of degree at most $n$ and $\mathcal{D}(\psi), \mathcal{D}(\varphi)$ are the Weyl-type differentiation operators. We also prove the Ulyanov and Kolyada-type inequalities in the Hardy spaces. Finally, we apply the obtained estimates to derive new embedding theorems for the Lipschitz and Besov spaces.