We present new formulas for decreasing rearrangements of functions and sequences in the spirit of Hardy and Littlewood in the context of convex functions. We apply them to introduce a new class of rearrangement invariant Banach spaces that are natural representations of dual spaces of Orlicz-Lorentz spaces.
(A work in collaboration with Anna Kaminska).

    The classical regularity theory is centered around two groups of results: (a) implicit (inverse) function and Lyusternik-Graves theorem and (b) Sard-Smale theorem and the transversality theory of Thom. It turns out that much of the theory can be productively extended as far as to set-valued mappings, in case (a) even between metric spaces. I shall briefly describe the basic ideas behind the extensions as well as some most principal results and applications.