Some properties of functions in Orlicz space

Abstract

For functions in Orlicz space L M * , we study the behavior of $$\mathop \smallint \nolimits_0^\tau x^*$$ (t)dt, where x* (t) is non-increasing and equimeasurable with ¦ x(t) ¦. We establish the existence of unbounded functions in L M * that are not limits of bounded functions and for which $$\mathop \smallint \nolimits_0^\tau x^* (t)dt = o(\tau M^{ - 1} (1/\tau ))$$ . Moreover, we establish a new criterion for an N-function to belong to the class Δ2 and a sufficiency test for a function to belong to Orlicz space.