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Title:  Lattice Properties and Interpolation Theory of the Spaces Lambda(psi,q) and M(psi) 
Author(s):  Lee, Chongsung 
Doctoral Committee Chair(s):  Peck, N. Tenney, 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  Since Lorentz introduced Lorentz space, there have been several generalizations of this space. Hunt and Cwikel studied Lorentz L$\sb{\rm p,q}$ spaces and showed some basic properties such as the characterization of the dual space of L$\sb{\rm p,q}$. Sharpley's version of Lorentz space is the space $\Lambda\sb\alpha$(X); he extended Calderon's interpolation theory of Lorentz L$\sb{\rm p,q}$ spaces to the spaces $\Lambda\sb\alpha$(X). In this thesis, we take Sharpley's Lorentz space $\Lambda\sb\alpha$(X) with minor modifications and define a Lorentz space $\Lambda\sb{\psi,{\rm q}}$. From its definition, it is easily observed that $\Lambda\sb{\psi,{\rm q}}$ is a symmetric space. Some geometrical properties of symmetric spaces are related to the growth rate of their fundamental functions which is always quasiconcave. We define the notion of ppower quasiconcavity to clarify this relation. We show that if the lower index of a given quasiconcave function $\psi(t)$ is strictly greater than zero, there exists p such that $\psi(t)$ is ppower quasiconcave. With the help of this notion, we extend some properties of Lorentz L$\sb{\rm p,q}$ space which were shown by Creekmore to the spaces $\Lambda\sb{\psi,{\rm q}}$. We also show the existence of bounded lattice isomorphisms from the Banach lattices $\ell\sb{\rm p}$, $\ell\sb\infty$ and L$\sb{\rm p}$ onto closed sublattices of Marcinkiewicz space. The well known Kmethod of Peetre allows us to construct interpolation spaces. One question is whether all interpolation spaces can be constructed by the Peetre Kmethod. Cwikel and Peetre showed that if a given Banach couples A is a Kmonotone space, all interpolation spaces can be constructed by the Peetre Kmethod. But, they really show only that all interpolation cones can be constructed by the Peetre Kmethod, rather than interpolation spaces; when they wrote their paper, an important result of Brudnyi and Krugljak was not available to them. We study this question when the given Banach couples A and B are different. In this case, we need a stronger condition, the strong $\lambda$Kmonotone property. We also show that every intermediate space A of the Banach couple A = ($\Lambda\sb{\varphi\sb0,1}, \Lambda\sb{\varphi\sb1,1}$) is a strong $\lambda$Kmonotone space with respect to A = ($\Lambda\sb{\varphi\sb0,1}, \Lambda\sb{\varphi\sb1,1}$) and B = (M$\sb{\psi\sb0}$,M$\sb{\psi\sb1}$). 
Issue Date:  1988 
Type:  Text 
Description:  98 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1988. 
URI:  http://hdl.handle.net/2142/71271 
Other Identifier(s):  (UMI)AAI8908741 
Date Available in IDEALS:  20141216 
Date Deposited:  1988 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois