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Title: Transference and the Hilbert transform on Banach function spaces
Author: McKain, David
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 2000
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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 well-known 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 Lp. 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 well-known functions spaces such as Lorentz spaces and weighted Lp spaces. The chapter finishes with the idea of vector-valued function spaces. In the second chapter we motivate and develop the idea of mixed (or iterated) norms, as introduced for Lp 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 Lp spaces. One rather technical highlight of this measure theory will be to make rigorous the canonical identification between most mixed norm spaces and vector-valued Banach function spaces. Motivated by a trivial application of Fubini's theorem which allows us to interchange two Lp 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 Lpq 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 Lp spaces to arbitrary rearrangement invariant spaces.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available