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

Newton–Besov spaces and Newton–Triebel–Lizorkin spaces on metric measure spaces In this paper, via a modification of the notion of weak upper gradients, we introduce and investigate properties of the Newton–Besov spaces $$\textit{NB}^s_{p,q}(X)$$ NB p , q s ( X ) and the Newton–Triebel–Lizorkin spaces $$\textit{NF}^s_{p,q}(X)$$ NF p , q s ( X ) , with $$s\in [0,1]$$ s ∈ [ 0 , 1 ] , $$1\le p<\infty $$ 1 ≤ p < ∞ and $$q\in (0,\infty ]$$ q ∈ ( 0 , ∞ ] , of functions on a metric measure space $$X$$ X and prove that, when $$1<p<\infty $$ 1 < p < ∞ , the space $$\textit{NB}^1_{p,\infty }(X)$$ NB p , ∞ 1 ( X ) coincides with the Newton–Sobolev space $$N^{1,p}(X)$$ 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 $$X={\mathbb R}^n$$ X = R n , these results are also new. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

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

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Publisher
Springer Journals
Copyright
Copyright © 2014 by Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-014-0291-7
Publisher site
See Article on Publisher Site

Abstract

In this paper, via a modification of the notion of weak upper gradients, we introduce and investigate properties of the Newton–Besov spaces $$\textit{NB}^s_{p,q}(X)$$ NB p , q s ( X ) and the Newton–Triebel–Lizorkin spaces $$\textit{NF}^s_{p,q}(X)$$ NF p , q s ( X ) , with $$s\in [0,1]$$ s ∈ [ 0 , 1 ] , $$1\le p<\infty $$ 1 ≤ p < ∞ and $$q\in (0,\infty ]$$ q ∈ ( 0 , ∞ ] , of functions on a metric measure space $$X$$ X and prove that, when $$1<p<\infty $$ 1 < p < ∞ , the space $$\textit{NB}^1_{p,\infty }(X)$$ NB p , ∞ 1 ( X ) coincides with the Newton–Sobolev space $$N^{1,p}(X)$$ 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 $$X={\mathbb R}^n$$ X = R n , these results are also new.

Journal

PositivitySpringer Journals

Published: Jun 8, 2014

References

  • On the well-posedness of the Euler equations in the Triebel–Lizorkin spaces
    Chae, D
  • Characterizations of Besov and Triebel–Lizorkin spaces on metric measure spaces
    Gogatishvili, A; Koskela, P; Zhou, Y

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