Matuszewska–Orlicz indices of the Sobolev conjugate Young function

Abstract

In this note we study the Matuszewska–Orlicz indices of Young and φ -functions and their conjugates. It is known, for example, that the index at zero of the inverse of a φ -function corresponds to the reciprocal of the index at infinity of the φ -function itself, and vice-versa . Likewise, the index at zero of the complementary Young function matches the Hölder conjugate of the index at infinity and the same holds for the opposite index. In this article we prove that the Matuszewska–Orlicz indices of the Sobolev conjugate Young function are equal to the Sobolev conjugate of the corresponding indices of the Young function. We then provide some examples as well which highlight the importance of an asymptotic condition in connection with Karamata theorem. Finally, we present a few examples and applications in the context of Orlicz spaces.