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. Positivity Springer Journals

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

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Springer Basel
Copyright © 2014 by Springer Basel
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
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