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.