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
Published: Jun 8, 2014
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