Some properties of the normal image of convex functions

Let z be a convex function defined in a convex domain D of a finite-dimensional Euclidean space. Denote by z ⁽ⁿ⁾ the convolutions of z with elements of a d-type sequence of test functions and let ? _z and ? _{z ⁽ⁿ⁾} be the measures of normal images corresponding to z and z ⁽ⁿ⁾. One of the main results of this work is that ? _{z ⁽ⁿ⁾}?? _z in variation on a compact K ? D if and only if ? _z is absolutely continuous on K with respect to Lebesgue measure. Bibliography: 9 titles.