On variation functions for subsequence ergodic averages

Authors
Citation
R. Nair et M. Weber, On variation functions for subsequence ergodic averages, MONATS MATH, 128(2), 1999, pp. 131-150
Citations number
12
Language
INGLESE
art.tipo
Article
Categorie Soggetti
Mathematics
Journal title
MONATSHEFTE FUR MATHEMATIK
ISSN journal
0026-9255 → ACNP
Volume
128
Issue
2
Year of publication
1999
Pages
131 - 150
Database
ISI
SICI code
0026-9255(1999)128:2<131:OVFFSE>2.0.ZU;2-U
Abstract
Suppose A(N)f = 1/N Sigma(k=1)(N)f(T(ak)x)(N = 1,2, ...) denote the ergodic averages for the natural numbers (a(k))(k=1)(infinity). Let Mf = sup(N gre ater than or equal to 1)/A(N)f/ denote the corresponding maximal function a nd let V(q)f = (Sigma(N=1)(infinity) /A(N+1)f - A(N)f/(q))(1/q) for q great er than or equal to 1. We show that for q, p > 1 if there exists C-p > 0 su ch that //Mf//(p) less than or equal to C-p//f//(p) then there exists C-p' > 0 such that //V(q)f//(p) less than or equal to C-p'//f//(p). Similar weak (1,1) inequalities follow for V-q when you know them for M too also with q > 1. We also show this fails completely if q = 1. We also show that for ce rtain polynomial like and random sequences (a(k))(k greater than or equal t o 1), if Sf = (Sigma(k=1)(infinity) /A(Nk+1)f - A(Nk)f/(2))(1/2), and (N-k)(k greater than or equal to 1) is of exponential growth then //Sf//(2) less than or equal to C//f//(2), for a certain positive constant C.