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arXiv · 1711.08163

Hardy-Littlewood and Ulyanov inequalities

Yurii Kolomoitsev·Sergey Tikhonov
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Abstract

We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness $ω_α(f,t)_q$ and $ω_β(f,t)_p$ for $0<p<q\le \infty$. A similar problem for the generalized $K$-functionals and their realizations between the couples $(L_p, W_p^ψ)$ and $(L_q, W_q^φ)$ is also solved. The main tool is the new Hardy-Littlewood-Nikol'skii inequalities. More precisely, we obtained the asymptotic behavior of the quantity $$ \sup_{T_n} \frac{\Vert \mathcal{D}(ψ)(T_n)\Vert_q}{\Vert \mathcal{D}(φ)(T_n)\Vert_p},\qquad 0<p<q\le \infty, $$ where the supremum is taken over all nontrivial trigonometric polynomials $T_n$ of degree at most $n$ and $\mathcal{D}(ψ), \mathcal{D}(φ)$ 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.

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Yurii Kolomoitsev, Sergey Tikhonov. 2017-11-22. Hardy-Littlewood and Ulyanov inequalities. https://arxiv.org/abs/1711.08163

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