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A Marcinkiewicz integral type characterization of the Sobolev space

Piotr Hajłasz and Zhuomin Liu

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In this paper we present a new characterization of the Sobolev space $W^{1,p}$, $1\lt p\lt \infty$ which is a higher dimensional version of a result of Waterman [32]. We also provide a new and simplified proof of a recent result of Alabern, Mateu, and Verdera [2]. Finally, we generalize the results to the case of weighted Sobolev spaces with respect to a Muckenhoupt weight.

Article information

Publ. Mat., Volume 61, Number 1 (2017), 83-104.

Received: 10 March 2015
First available in Project Euclid: 22 December 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems
Secondary: 42B25: Maximal functions, Littlewood-Paley theory

Sobolev spaces Littlewood--Paley theory


Hajłasz, Piotr; Liu, Zhuomin. A Marcinkiewicz integral type characterization of the Sobolev space. Publ. Mat. 61 (2017), no. 1, 83--104. doi:10.5565/PUBLMAT_61117_03.

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