On the comparison of the spaces \(L^1 BV({\mathbb R}^n)\) and \(BV({\mathbb R}^n)\) (Q2758976)
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scientific article; zbMATH DE number 1680636
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the comparison of the spaces \(L^1 BV({\mathbb R}^n)\) and \(BV({\mathbb R}^n)\) |
scientific article; zbMATH DE number 1680636 |
Statements
10 December 2001
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\(L^1\)-variation
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variation
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Favard class
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On the comparison of the spaces \(L^1 BV({\mathbb R}^n)\) and \(BV({\mathbb R}^n)\) (English)
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The main result is NEWLINE\[NEWLINE\bigl\{f\in L^1:L^1 \text{Var}(f) <\infty \bigr\}= \bigl\{f\in L^1:\text{Var}(f)< \infty\bigr\},NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\text{Var}(f): =\sup \Bigl\{\int_{\mathbb{R}^n} f \text{div} \Phi dx:\Phi\in C^1_0(\mathbb{R}, \mathbb{R}), |\Phi|\leq 1\Bigr\}NEWLINE\]NEWLINE and NEWLINE\[NEWLINEL^1\text{Var}(f):=\sup \left\{1/ |h |\int_{\mathbb{R}^n} \bigl|f(x+h)-f(x) \bigr|dx:h\in\mathbb{R}^n \setminus \{0\} \right\}.NEWLINE\]
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