A Lindenstrauss theorem for some classes of multilinear mappings

Under some natural hypotheses, we show that if a multilinear mapping belongs to some Banach multlinear ideal, then it can be approximated by multilinear mappings belonging to the same ideal whose Arens extensions simultaneously attain their norms. We also consider the class of symmetric multilinear mappings.Comment: 11 page

Quantitative estimates and extrapolation for multilinear weight classes

In this paper we prove a quantitative multilinear limited range extrapolation theorem which allows us to extrapolate from weighted estimates that include the cases where some of the exponents are infinite. This extends the recent extrapolation result of Li, Martell, and Ombrosi. We also obtain vector-valued estimates including $\ell^\infty$ spaces and, in particular, we are able to reprove all the vector-valued bounds for the bilinear Hilbert transform obtained through the helicoidal method of Benea and Muscalu. Moreover, our result is quantitative and, in particular, allows us to extend quantitative estimates obtained from sparse domination in the Banach space setting to the quasi-Banach space setting. Our proof does not rely on any off-diagonal extrapolation results and we develop a multilinear version of the Rubio de Francia algorithm adapted to the multisublinear Hardy-Littlewood maximal operator. As a corollary, we obtain multilinear extrapolation results for some upper and lower endpoints estimates in weak-type and BMO spaces.Comment: 44 pages. Minor improvements. To appear in Mathematische Annale

An abstract proximal point algorithm

The proximal point algorithm is a widely used tool for solving a variety of convex optimization problems such as finding zeros of maximally monotone operators, fixed points of nonexpansive mappings, as well as minimizing convex functions. The algorithm works by applying successively so-called "resolvent" mappings associated to the original object that one aims to optimize. In this paper we abstract from the corresponding resolvents employed in these problems the natural notion of jointly firmly nonexpansive families of mappings. This leads to a streamlined method of proving weak convergence of this class of algorithms in the context of complete CAT(0) spaces (and hence also in Hilbert spaces). In addition, we consider the notion of uniform firm nonexpansivity in order to similarly provide a unified presentation of a case where the algorithm converges strongly. Methods which stem from proof mining, an applied subfield of logic, yield in this situation computable and low-complexity rates of convergence