The purpose of this paper is
to obtain an Lp estimate for the supremum of the Cesàro averages of a
certain class of positive contractions of Lp. Let (X,ℱ,μ) be a measure space,
and let T be a linear operator mapping Lp(X,ℱ,μ) into itself for p fixed,
1 < p < +∞. If there is a constant c > 0 such that for each f ∈ Lp(X,ℱ,μ),
then
we say that T admits of a dominated estimate with constant 0. In an effort to unify
certain results due to A. IonescuTulcea and to E. Stein, a somewhat more general
form of the following theorem was obtained earlier: If T is a positive contraction, and
if there exists an h > 0 a.e., h ∈ Lp(X,ℱ,μ) and Th = h, then T admits of a
dominated estimate with constant p∕p − 1. In the present paper, we have extended
the theorem, obtaining a slightly more general form of the following: If T is a positive
contraction and if for each positive integer n there exists an hn> 0 a.e.,
hn∈ Lp(X,ℱ,μ) and ∥hn∥ = ∥Tnhn∥, then T admits of a dominated estimate with
constant p∕p − 1.
