# 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

, Volume 19 (2) – Jun 8, 2014
44 pages

/lp/springer_journal/newton-besov-spaces-and-newton-triebel-lizorkin-spaces-on-metric-yNdjIwyoS3
Publisher
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

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Unlimited reading Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere. ### Stay up to date Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates. ### Organize your research It’s easy to organize your research with our built-in tools. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. ### DeepDyve Freelancer ### DeepDyve Pro Price FREE$49/month

\$360/year
Save searches from