Let \(\mu\),\(\nu\) be two measures on \(R^ n\). Suppose that a function \(\phi_ Q\) supported in Q is associated with each cube Q in \(R^ n\). Let \(Mf(x)=\sup \int f\phi_ Qd\nu\) be the maximal operator where the sup is extended over all Q with center x. In the previous paper [Trans. Am. Math. Soc. 275, 821-831 (1983; Zbl 0509.42023)] the authors have established the inequality \[ (1)\quad(Mf)^*\!_{\mu}(\xi(\leq A\int^{\infty}_{0}\Phi(t)f^*\!_{\nu}(t\xi)dt, \] where \(f^*\!_{\nu}\) is the non-decreasing rearrangement of f with respect to the measure \(\nu\), and \(\Phi(t)=\sup_{Q}\{\mu(Q)\phi^*\!_{Q,\nu}(\mu(Q)t)\}.\) In this paper the authors investigate the iteration \(M_ j\) of the Hardy- Littlewood maximal function \(Mf(x)=\sup \frac{1}{| Q|}\int_{Q}f(t)dt.\)
The main result is as follows: Let \(d\mu =udx\), \(d\nu =vdx\) where (u,v) is a pair of weights with \(u\geq 0\) in \(L^ 1\!_{loc}(R^ n)\) and \(0<v<\infty\) a.e. If \(\| Mf\|_{q,u}\leq B_ p\| f\|_{q,v}, 1\leq p<q\), then for each \(q>p\) there is a constant \(0<A_ q<\infty\) such that \(\| M_ jf\|_{q,u}\leq A^ j\!_ q\| f\|_{q,v}.\)
This theorem implies that extrapolation of \(\| Mf\|_{p,u}\leq B\| f\|_{p,v},\) i.e. \(\| Mf\|_{p-\epsilon,u}\leq B\| f\|_{p-\epsilon,v}\) for some \(\epsilon>0\), is possible if and only if \(\| M_ j\| =O(A^ j)\) as \(j\to \infty\). The authors also investigate how (1) can be used to study the restricted weak type behavior of a general maximal operator. Finally, the generalizations of (1) to abstract measure spaces are presented.