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Littlewood-Paley Characterizations of Second-Order Sobolev Spaces via Averages on Balls

  Published:2015-08-17
 Printed: Mar 2016
  • Ziyi He,
    School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People's Republic of China
  • Dachun Yang,
    School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People's Republic of China
  • Wen Yuan,
    School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People's Republic of China
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Abstract

In this paper, the authors characterize second-order Sobolev spaces $W^{2,p}({\mathbb R}^n)$, with $p\in [2,\infty)$ and $n\in\mathbb N$ or $p\in (1,2)$ and $n\in\{1,2,3\}$, via the Lusin area function and the Littlewood-Paley $g_\lambda^\ast$-function in terms of ball means.
Keywords: Sobolev space, ball means, Lusin-area function, $g_\lambda^*$-function Sobolev space, ball means, Lusin-area function, $g_\lambda^*$-function
MSC Classifications: 46E35, 42B25, 42B20, 42B35 show english descriptions Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
Maximal functions, Littlewood-Paley theory
Singular and oscillatory integrals (Calderon-Zygmund, etc.)
Function spaces arising in harmonic analysis
46E35 - Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
42B25 - Maximal functions, Littlewood-Paley theory
42B20 - Singular and oscillatory integrals (Calderon-Zygmund, etc.)
42B35 - Function spaces arising in harmonic analysis
 

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