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

We prove the existence and Gevrey regularity of local solutions of the three dimensional periodic Navier-Stokes equations in case the sequence of Fourier coefficients of the initial data lies in an appropriate weighted p space. Our work is motivated by that of Foias and Temam ([10]) and we obtain a generalization of their result. In particular, our analysis allows for initial data that are less smooth than in [10] and can also have infinite energy. Our main tool is a variant of the Young convolution inequality enabling us to estimate the nonlinear term in weighted p norm.

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Abstract. For functions whose derivatives belong to an Orlicz space, we develop their "fine" properties as a generalization of the treatment found in [MZ] for Sobolev functions. Of particular importance is Theorem 8.8, which is used in the development in [MSZ] of the coarea formula for such functions.