Topics in the <em>L^p</em> theory for outer measure spaces

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 to the study of the boundedness properties of some multilinear forms satisfying certain invariances arising in the context of Calderón-Zygmund theory and time-frequency analysis. <br /> However, the theory can be developed in a broader generality of settings. It requires a set X, an outer measure μ to evaluate the magnitude of subsets of X, and a size S to evaluate the magnitude of functions on X when localized to the elements of a certain collection A of subsets of X. Then, the outer L^p_μ(S) quasi-norms are defined by the interplay between μ and S via a layer cake integral. For example, the mixed L^p spaces on the Cartesian product of σ-finite measure spaces can be exhibited as outer L^p spaces for an appropriate choice of (X,μ,S). <br /> Do and Thiele developed the theory of outer L^p spaces in the direction of their real interpolation properties, such as Hölder's inequality and Marcinkiewicz interpolation. This thesis is concerned with further developing the theory of these spaces. The focus is towards the Banach space properties analogous to those of the mixed L^p spaces, such as Köthe duality, triangle inequality for countably many summands, and Minkowski's inequality. <br /> The thesis consists of four chapters. <br /> Chapter 1 is an introduction. We recall definitions and properties of outer L^p spaces from the article of Do and Thiele and we introduce a list of examples. We also comment on the results about the Banach space properties of outer L^p spaces appearing in the following chapters. <br /> In Chapter 2, we study single iterated outer L^p spaces, when the size is a suitably averaged local classical L^r quasi-norm associated with a measure ω on X. For p and r in (1,∞) we prove that the outer L p quasi-norms are equivalent to norms up to a constant uniform in the setting (X,μ,ω). We also focus on the setting on R^d × (0,∞) associated with Calderón-Zygmund theory. <br /> In Chapter 3, we study double iterated outer L^p spaces, when the size is a suitably averaged local single iterated outer L^q quasi-norm on the setting (X,ν,ω). Under additional assumptions on μ and ν, for p, q, and r in (1,∞) we prove that the outer L^p quasi-norms are equivalent to norms up to a constant uniform in the setting (X,μ,ν,ω). We provide counterexamples showing the necessity of additional assumptions. We also focus on the setting on R² × (0,∞) associated with time-frequency analysis. <br /> In Chapter 4, we address additional questions about outer L^p spaces. For example, we prove a version of Minkowski's inequality for single iterated outer L^p quasi-norms. We conclude with some open conjectures.