Rubio de Francia's extrapolation theory: estimates for the distribution function (Q2880396)

The authors prove the weak type version of Rubio de Francia's extrapolation results and many others, including new boundedness properties of operators on different kinds of spaces.NEWLINENEWLINEWe shall use the Hardy-Littlewood maximal operator NEWLINE\[NEWLINE Mf(x)=\sup_{x \in Q}\frac{1}{|Q|}\int_{Q}|f(x)|dy, NEWLINE\]NEWLINE where the supremum is taken over all cubes \(Q \ni x\).NEWLINENEWLINELet \(T\) be an arbitrary operator bounded from \(L^{p_{0}}(w)\) into \(L^{p_{0},\infty}(w)\) for every weight \(w\) in the Muckenhoupt class \(A_{p_{0}}\) with \(1 < p_{0} < \infty\). Then it is proved that the distribution function of \(Tf\) with respect to any weight \(u\) can be essentially majorized by the distribution function of \(Mf\) with respect to \(u\) (plus an integral term easy to control). As a consequence, extrapolation results, including results in a multilinear setting, can be obtained with very simple proofs.NEWLINENEWLINENew applications in extrapolation for two-weight problems and estimates on rearrangement invarinat spaces are established, too.