Hardy–Littlewood and Ulyanov inequalities

Abstract

We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness ω α ( f , t ) q \omega _\alpha (f,t)_q and ω β ( f , t ) p \omega _\beta (f,t)_p for 0 > p > q ≤ ∞ 0>p>q\le \infty . A similar problem for the generalized K K -functionals and their realizations between the couples ( L p , W p ψ ) (L_p, W_p^\psi ) and ( L q , W q φ ) (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 ‖ D ( ψ ) ( T n ) ‖ q ‖ D ( φ ) ( T n ) ‖ p , 0 > p > q ≤ ∞ , \begin{equation*} \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 , \end{equation*} where the supremum is taken over all nontrivial trigonometric polynomials T n T_n of degree at most n n and D ( ψ ) , D ( φ ) \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.