Necessary and sufficient conditions for the chain rule in $W_{\mathrm{loc}}^{1,1}(\mathbb{R}^N; \mathbb{R})$ and $BV_{\mathrm{loc}}(\mathbb{R}^N; \mathbb{R}^d)$

Giovanni Leoni; Massimiliano Morini

doi:10.4171/jems/78

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Necessary and sufficient conditions for the chain rule in $W_{\mathrm{loc}}^{1,1}(\mathbb{R}^N; \mathbb{R})$ and $BV_{\mathrm{loc}}(\mathbb{R}^N; \mathbb{R}^d)$

Giovanni Leoni, Massimiliano Morini

Journal of the European Mathematical Society2007doi:10.4171/jems/78

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Abstract

In this paper we prove necessary and sufficient conditions for the validity of the classical chain rule in Sobolev spaces and in the space of functions of bounded variation.

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Giovanni Leoni#1 Massimiliano Morini#2

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