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

Necessary and sufficient conditions for the chain rule in $W_{loc}^{1, 1} (R^{N}; R)$ and $B V_{loc} (R^{N}; R^{d})$

Giovanni Leoni
Carnegie-Mellon University, Pittsburgh, USA
Massimiliano Morini
Carnegie Mellon University, Pittsburgh, United States

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

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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.

Cite this article

Giovanni Leoni, Massimiliano Morini, Necessary and sufficient conditions for the chain rule in $W_{loc}^{1, 1} (R^{N}; R)$ and $B V_{loc} (R^{N}; R^{d})$ . J. Eur. Math. Soc. 9 (2007), no. 2, pp. 219–252

DOI 10.4171/JEMS/78