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.
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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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Nageswari Shanmugalingam is partially supported by Grant No. 200474 from the Simons Foundation. Dachun Yang is supported by the National Natural Science Foundation of China (Grant Nos. 11171027 and 11361020). Wen Yuan is supported by the National Natural Science Foundation of China (Grant No. 11101038). This project is also partially supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20120003110003) and the Fundamental Research Funds for Central Universities of China (Grant Nos. 2012LYB26 and 2012CXQT09).
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Shanmugalingam, N., Yang, D. & Yuan, W. Newton–Besov spaces and Newton–Triebel–Lizorkin spaces on metric measure spaces. Positivity 19, 177–220 (2015). https://doi.org/10.1007/s11117-014-0291-7
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DOI: https://doi.org/10.1007/s11117-014-0291-7
Keywords
- Newton–Besov space
- Newton–Triebel–Lizorkin space
- Newton–Sobolev space
- Upper gradient
- Hajłasz gradient
- Modulus
- Poincaré inequality
- Semmes pencil of curves