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Hardy–Littlewood and Ulyanov inequalities

About this Title

Yurii Kolomoitsev and Sergey Tikhonov

Publication: Memoirs of the American Mathematical Society
Publication Year: 2021; Volume 271, Number 1325
ISBNs: 978-1-4704-4758-8 (print); 978-1-4704-6628-2 (online)
DOI: https://doi.org/10.1090/memo/1325
Published electronically: July 6, 2021
Keywords: Moduli of smoothness, $K$-functionals, Ulyanov’s inequalities, Hardy–Litlewood–Nikol’skii’s inequalities, embedding theorems, fractional derivatives

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Table of Contents

Chapters

  • Basic notation
  • 1. Introduction
  • 2. Auxiliary results
  • 3. Polynomial inequalities of Nikol’skii–Stechkin–Boas–types
  • 4. Basic properties of fractional moduli of smoothness
  • 5. Hardy–Littlewood–Nikol’skii inequalities for trigonometric polynomials
  • 6. General form of the Ulyanov inequality for moduli of smoothness, $K$-functionals, and their realizations
  • 7. Sharp Ulyanov inequalities for $K$-functionals and realizations
  • 8. Sharp Ulyanov inequalities for moduli of smoothness
  • 9. Sharp Ulyanov inequalities for realizations of $K$-functionals related to Riesz derivatives and corresponding moduli of smoothness
  • 10. Sharp Ulyanov inequality via Marchaud inequality
  • 11. Sharp Ulyanov and Kolyada inequalities in Hardy spaces
  • 12. $(L_p,L_q)$ inequalities of Ulyanov-type involving derivatives
  • 13. Embedding theorems for function spaces
  • List of symbols

Abstract

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 \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 $T_n$ 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.

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