New extrapolation estimates (Q1579197)

In this paper, the author gives several types of extrapolation results. One of them allows the author to improve Yano's extrapolation theorem. Given a sublinear operator \(T\) satisfying \[ \|T\chi_A\|_{L^p(\nu)}\leq \frac{C}{p-1}\|\chi_A\|_{L^p(\mu)}, \] for every measurable set \(A\) and every \(1<p\leq p_0\), with \(C\) independent of \(A\) and \(p\), we show that \[ \sup_{r>0}\frac{\int^\infty_{1/r}\lambda^\nu_{Tf}(y)\,dy}{1+\log^+ r}\lesssim\int_{\mathcal M}|f(x)|(1+\log^+|f(x)|)\,d\mu(x). \]