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Title: Isomorphic classification of weak Lp spaces
Keywords: weak Lp space, non-atomic measure space, embedding
Issue Date: 9-Jan-2007
Citation: RUDY SABARUDIN (2007-01-09). Isomorphic classification of weak Lp spaces. ScholarBank@NUS Repository.
Abstract: Leung (1996) studied isomorphic classification of purely atomic weak Lp spaces. He showed that these spaces can be classified into five mutually exclusive isomorphic classes. We would like to extend this result and continue the research to obtain isomorphic classification of purely non-atomic weak Lp spaces.The thesis contains four chapters, including the introduction chapter. Using results from Carothers and Dilworth (1985-1986 and 1988), we start with the observation on the subspace structure of L^{p,a?z}({-1,1}^I) for uncountable I. It will be shown that L^{p,a?z}(\oplus_{I} [0,1]) can be embedded into L^{p,a?z}({-1,1}^I) iff 1 < p < 2.Next, we use Maharama??s theorem to obtain standard representation of purely non-atomic weak Lp spaces and observe some properties of the embeddings between weak Lp spaces L^{p,a?z}(\oplus_{I? \in A} a_I?{-1,1}^{I_I?}). We will introduce three isomorphic-invariant parameters in terms of |A| and sup |I_I?| and obtain the characterization of embeddings between certain weak Lp spaces in terms of I_I?. These parameters and characterization are useful to study the uniqueness of the classification.Finally, the thesis concludes with the classification theorem: for purely non-atomic measure spaces (I?,I#,I?), L^{p,a?z}(I?,I#,I?), pa?Y2, can be classified into two mutually exclusive groups, each containing nine isomorphic classes.
Appears in Collections:Ph.D Theses (Open)

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