The co-area formula for Sobolev mappings

We extend Federer’s co-area formula to mappings f f belonging to the Sobolev class W 1 , p ( R n ; R m ) W^{1,p}(\mathbb {R}^n;\mathbb {R}^m) , 1 ≤ m > n 1 \le m > n , p > m p>m , and more generally, to mappings with gradient in the Lorentz space L m , 1 ( R n ) L^{m,1}(\mathbb {R}^n) . This is accomplished by showing that the graph of f f in R n + m \mathbb {R}^{n+m} is a Hausdorff n n -rectifiable set.