Weighted inequalities for the Hardy operator

Abstract

This project revolves around Hardy’s integral inequality, proved by G. H. Hardy in 1925. This inequality has been studied by a large number of authors during the twentieth century and has motivated some important lines of study which are currently active. We study the classical Hardy’s integral inequality and its generalizations. We analyse some of the first results including weighted inequalities and prove the key theorem of B. Muckenhoupt, who characterized Hardy’s integral inequality with weights for the diagonal case in 1972. After this fundamental result, different authors considered the general context and new characterizations appeared until closing definitely the problem in 2000. Also we study Hardy’s integral inequality in the cone of monotone functions. This point of view is really interesting and has a lot of surprising consequences. For example, M. A. Ariño and B. Muckenhoupt realized in 1990 that Hardy’s inequality in the cone of monotone functions is equivalent to the boundedness of the Hardy-Littlewood maximal operator between Lorentz spaces. Just after E. Sawyer proved that the classical Lorentz space $\Lambda ^{p}(w)$ is normable if, and only if, Hardy’s integral inequality in the cone of monotone functions is satisfied for $w$. We study also the normability of both spaces $\Lambda^{p} (w)$ and $\Lambda^{p,\infty} (w)$ in terms of the boundedness of the maximal operator.