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Positivity

, Volume 19, Issue 2, pp 177–220 | Cite as

Newton–Besov spaces and Newton–Triebel–Lizorkin spaces on metric measure spaces

  • Nageswari Shanmugalingam
  • Dachun YangEmail author
  • Wen Yuan
Article

Abstract

In this paper, via a modification of the notion of weak upper gradients, we introduce and investigate properties of the Newton–Besov spaces \(\textit{NB}^s_{p,q}(X)\) and the Newton–Triebel–Lizorkin spaces \(\textit{NF}^s_{p,q}(X)\), with \(s\in [0,1]\), \(1\le p<\infty \) and \(q\in (0,\infty ]\), of functions on a metric measure space \(X\) and prove that, when \(1<p<\infty \), the space \(\textit{NB}^1_{p,\infty }(X)\) coincides with the Newton–Sobolev space \(N^{1,p}(X)\). A Poincaré type inequality related to these function spaces is also investigated. Sensitivity to changes of functions in these classes on sets of measure zero is also demonstrated. Even in the Euclidean setting \(X={\mathbb R}^n\), these results are also new.

Keywords

Newton–Besov space Newton–Triebel–Lizorkin space  Newton–Sobolev space Upper gradient Hajłasz gradient Modulus  Poincaré inequality Semmes pencil of curves 

Mathematics Subject Classification (2010)

Primary 42B35 Secondary 46E35 30L99 

Notes

Acknowledgments

The authors would like to thank the referee for her/his very careful reading and helpful comments which improve the presentation of this article.

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Nageswari Shanmugalingam
    • 1
  • Dachun Yang
    • 2
    Email author
  • Wen Yuan
    • 2
  1. 1.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA
  2. 2.School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex SystemsMinistry of EducationBeijingPeople’s Republic of China

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