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 $0<p<q\le \mathrm{\infty }$. 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 $$\underset{T_n}{\sup }\frac{\mathrm{\parallel }\mathcal{D}(\psi )(T_n){\mathrm{\parallel }}_q}{\mathrm{\parallel }\mathcal{D}(\phi )(T_n){\mathrm{\parallel }}_p},0<p<q\le \mathrm{\infty },$$ where the supremum is taken over all nontrivial trigonometric polynomials $T_n$ of degree at most $n$ and $\mathcal{D}(\psi ),\mathcal{D}(\phi )$ 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.