On the differentiability of functions which are of bounded variation in Tonelli's sense

where the integral is extended over En. By (j, g) we mean f f g dx, assuming that the integral converges absolutely, and by f * g the convolution of f and g. We shall also consider completely additive functions [J. (E) of Borel subsets E of En. By (j, [J.) we mean f f d [J. and by f * [J. the convolution f f (x y) d [J. (y). By D we denote the class of infinitely differentiable functions in En with compact support. Finally, C win stand for a constant depending only on the dimension and the parameters displayed. A locally integrable function is said to have first derivatives hex), j = 1, 2, ... , n, in the sense of distributions if (cp, h) = (~, f) a Xj