A general chain rule for distributional derivatives

Abstract

We prove a general chain rule for the distribution derivatives of the composite function υ ( x ) = f ( u ( x ) ) \upsilon (x) = f(u(x)) , where u : R n → R m u:{{\mathbf {R}}^n} \to {{\mathbf {R}}^m} has bounded variation and f : R m → R k f:{{\mathbf {R}}^m} \to {{\mathbf {R}}^k} is Lipschitz continuous.