Weight characterization of an averaging operator.
Let \(0<\alpha<1\) and let \[ (T_{\alpha}f)(x):=\frac1{(1-\alpha)x}\left(\int_{\alpha x}^ xf\right),\qquad x\geq0, \] for \(f\) measurable on \([0,\infty)\). When \(\alpha=0\), then this operator coincides formally with the Hardy averaging operator. If \(\alpha\neq0\), then the averages are calculated over a moving interval of variable length. Such operators were studied, and their boundedness on weighted Lebesgue spaces was characterized, by \textit{H. P. Heinig} and \textit{G. Sinnamon} [Stud. Math. 129, No.~2, 157--177 (1998; Zbl 0910.26008)] and, in a more general setting, by \textit{A. Gogatishvili} and \textit{J. Lang} [J. Inequal. Appl. 4, No.~1, 469--471 (1999; Zbl 0947.47020)]. (Let us recall that the weighted Lebesgue space \(L^ p(w)\) is, for \(1\leq p<\infty\) and \(w\) a nonnegative weight, determined by the norm \((\int| f| ^ pw)^{1/p}\).) The main result of the paper under review is the factorization theorem, stating that, for \(0<\alpha<1\), \(1<p<\infty\), \(f\geq0\) and for a pair of positive weights \(v,w\), \(T_\alpha f\in L^ p(w)\) if and only if \(f\) admits the factorization \[ f=gh,\quad g\geq0,\;h>0\quad \text{on}\;[0,\infty), \] such that \(g\in L^ p(v)\) and \[ \sup_{\{[x,t];\;0<t\leq x<t/\alpha\}} \left(\int_t^ x\frac{w(s)}{s^ p}\,ds\right)^{1/p} \left(\int_{\alpha x}^ tv^{1-p'}h^{p'}\,dt\right)^{1/p'}<\infty. \] As a~consequence, a~two-sided estimate for \(\| T_{\alpha} f\| _{L^ p(w)}\) is obtained. The main result extends also to the case \(\alpha=0\), covering thereby an earlier factorization result of \textit{G. Bennett} [Mem. Am. Math. Soc. 576 (1996; Zbl 0857.26009)]. Finally, norm estimates comparing the operator \(T_0\) to its dual are established.
scientific article; zbMATH DE number 1970522
scientific article