Functions of bounded higher variation

Abstract

We say that a function u : R-m --> R-n, with m greater than or equal to n, has bounded n-variation if Der (u(xalpha1), . . . , u(xalphan)) is a measure for every 1 less than or equal to alpha(1) < (. . .) < alpha(n) less than or equal to m. Here Det(v1, . . . , v(n)) denotes the distributional determinant of the matrix whose columns are the given vectors, arranged in the given order. In this paper we establish a number of properties of BnV functions and related functions. We establish general (and rather weak) versions of the chain rule and the coarea formula; we show that stronger forms of the chain rule can fail, and we also demonstrate that BnV functions cannot, in general, be strongly approximated by smooth functions; and we prove that if u e BnV(R-m, R-n) and \u\ = 1 a.e., then the jacobian of u is an m - n-dimensional rectifiable current.