Vol. 30, No. 3, 1969

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Estimates of positive contractions

Rafael Van Severen Chacon and Stephen Allan McGrath

Vol. 30 (1969), No. 3, 609–620

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,) be a measure space, and let T be a linear operator mapping Lp(X,) into itself for p fixed, 1 < p < +. If there is a constant c > 0 such that for each f Lp(X,),

n        p
sunp|f,(f + Tf )∕2,⋅⋅⋅ ,(f + Tf + ⋅⋅⋅+ T f)∕n+ 1| dμ
p∫    p
≦ c   |f|d.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,) 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,) and hn= Tnhn, then T admits of a dominated estimate with constant p∕p 1.

Mathematical Subject Classification
Primary: 28.70
Secondary: 47.00
Received: 17 January 1969
Published: 1 September 1969
Rafael Van Severen Chacon
Stephen Allan McGrath