Two weight extrapolation via the maximal operator
There are obtained extrapolation theorems for weighted \(L^p\)-spaces with pairs of weights \((w,M^kw)\) and \((w,(Mw/w)^rw)\), where \(M^k\) is the \(k\)th iterate of the Hardy-Littlewood maximal operator. As an example, I shall quote Theorem 1.1.1: Let \(\|\cdot,L^p_w\|\) denote the norm in the weighted space \(L^p_w(\mathbb{R}^n)\). Let \(S\) and \(T\) be operators (not necessarily linear) and let \(f\) be a function in a suitable test class for both \(S\) and \(T\). Let us suppose, there are constants \(p_0\) and \(C_0\) and a positive integer \(k\) such that for all weights \(w\) there holds \(\|Tf,L^{p_0}_w\|\leq C_0\|Sf, L^{p_0}_{M^kw}\|\). Then for every \(p\in (p_0,\infty)\) there exists a constant \(C_p\) (dependent on \(p_0\), \(p\), \(k\), \(n\)) such that the inequality \(\|Tf, L^p_w\|\leq C_p\|Sf, L^p_{M^{[kp/p_0]+ 1_w}}\|\) holds for all weights \(w\). There are given applications to square functions and to Calderón-Zygmund singular integral operator.
Calderón-Zygmund singular integral operator
scientific article; zbMATH DE number 1502200
scientific article