Title:

Transference and the Hilbert transform on Banach function spaces

The thesis begins with a summary of the classical theory of Banach function spaces, including the notions of saturation and associate norms along with the various wellknown ideas of completeness. We then go on to establish some rather more "practical" results. In particular we look at the problem of establishing when certain subspaces, such as the simple or continuous functions, are dense in a Banach function space L_{p}. We shall see that our intention fails us slightly when considering continuous functions, requiring us to approach that idea of saturation from a slightly different angle. This interplay between topology, measure and norm is studied in more depth when we look at function norms over locally compact abelian groups, and results will be illuminated by reviewing wellknown functions spaces such as Lorentz spaces and weighted L^{p} spaces. The chapter finishes with the idea of vectorvalued function spaces. In the second chapter we motivate and develop the idea of mixed (or iterated) norms, as introduced for L^{p} spaces by Benedek and Panzone, before going on to identify dense subspaces and some other elementary results. We shall see that there are certain interesting measurability problems to address here which are not evident when considering L^{p} spaces. One rather technical highlight of this measure theory will be to make rigorous the canonical identification between most mixed norm spaces and vectorvalued Banach function spaces. Motivated by a trivial application of Fubini's theorem which allows us to interchange two L^{p} norms, i.e. f(x, y)_{Lp(dy)}_{Lp(dx)} = f(x, y)_{Lp(dx)}_{Lp(dy)}, we then consider when interchanging two general mixed norms is bounded. Although there are some positive results we shall see that this idea fails in many cases. In particular we shall show that two iterated Lorentz L^{pq} norms can be interchanged if and only if p = q. In chapter three we study how the classical transference theorem of Coifman and Weiss can be generalised from L^{p} spaces to arbitrary rearrangement invariant spaces.
