Some Inequalities for the Mean of Almost Periodic Measures

Silvia - Otilia Corduneanu
Department of Mathematics
Gh. Asachi Technical University, Iasi
Bd. Copou 11
6600 Iasi, ROMANIA.

Abstract: J. Lamadrid and L. Argabright defined the almost periodic measures on a locally compact abelian group $G$ and the mean of theirs. They proved that the measure $f\mu$ which is defined by an almost periodic function $f$ as density and an almost periodic measure $\mu$ as base, is also an almost periodic measure.

In our paper we shall explore the following theme: given an almost periodic function $f$, an almost periodic measure $\mu$ satisfying certain conditions and $c \in I\!\!R$, determine when the inequality

$$f(x) \leq c + M_y[f(xy^{-1})\mu(y)], \;\;x \in G,$$

implies the existence of a positive constant $k$ such that

$$M[\frac{1}{f}\mu] \geq k.$$

We denote by $M[\frac{1}{f}\mu]$ the mean of the almost periodic measure defined by $\frac{1}{f}$ as density and by $\mu$ as base. For $x \in G$ we denote by $M_y[f(xy^{-1})\mu(y)]$ the mean of the almost periodic measure $g\mu$, where $g(y) = f(xy^{-1}), \forall y \in G$.