On the differentiability of functions in ${\bf R}\sp{n}$

Abstract

It can be shown [3, Chapter V, §6.3] that the space Ln cannot be replaced by any larger Lorentz space and in this sense the theorem is sharp. In this note, we shall prove Stein's Theorem using elementary principles. This should be compared with the proofs [2 and 1] which use more sophisticated Fourier analytic methods. Since this is a local theorem, we may assume that O = Qr, is a cube in Rn. Recall that a function yj is in Ln if and only if its decreasing rearrangement tp* (see [4, p. 189]) satisfies