Reverse inequalities
Let \((X,\beta,m)\) be a \(\sigma\)-finite measure space, and let \((T_n)\) be a sequence of operators from \(L_2(X)\) to \(L_2(X)\). The maximal operator for \((T_n)\) is defined by \[ T^{\sharp} f=\sup_n|T_nf|. \] Typically, a strong maximal inequality is an inequality of the form \(\|T^{\sharp} f\|_2\leq C\|f\|_2\), where \(C>0\) is some constant independent of \(f\in L_2(X)\). This paper is devoted to an interesting study of ``reverse'' maximal inequalities, i.e., inequalities of the form: There exists \(C>0\) with \(\|f\|_2\leq C\|T^{\sharp} f\|_2\) for all \(f\). The authors start studying differentiation operators. They give positive and negative results. Then they go on to look at ergodic averages. Here \(T_k\) is of the form \(A_{n_k}\), where \[ A_n f(x)= n^{-1} \sum^{n-1}_{i=0} f(\tau^i x) \] and \(\tau\) is an invertible measure preserving transformation. An example in the case of ergodic square functions is the following: Let \((n_k)\) be a lacunary sequence with no nontrivial common divisor. For any \(p\) with \(1< p<\infty\) there is a \(C_p>0\) such that for all \(f\in L_p(X)\) \[ \|f\|_p\leq C_p\Biggl\|\Biggl( \sum^\infty_{k= 1}|A_{n_k} f- A_{n_{k+1}} f|^2\Biggr)^{1/2}\Biggr\|_p. \]
scientific article; zbMATH DE number 1445615