Analysis and Probability Seminar

Abstract: The theory of L^p spaces for outer measures, or outer L^p spaces, was introduced by Do and Thiele. Their main interest was its application in developing a unifying framework for the estimate of some invariant multilinear forms arising in the context of harmonic analysis (Calderón-Zygmund theory, time-frequency analysis). To this end, they developed the theory in the direction of the interpolation properties of the spaces, such as Hölder's inequality and Marcinkiewicz interpolation.

However, the outer L^p spaces can be defined in a broader generality of settings, for example extending the classical notion of mixed L^p spaces on the Cartesian product of sigma-finite measure spaces. In this talk we expose further developments in the theory of the outer L^p spaces. The focus is towards their Banach space properties analogous to those of the mixed L^p spaces, such as Fubini's theorem, Köthe duality, triangle inequality, and Minkowski's inequality.