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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
Degree Granting Institution:University of Illinois at Urbana-Champaign
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 p-power 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 p-power 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 K-method of Peetre allows us to construct interpolation spaces. One question is whether all interpolation spaces can be constructed by the Peetre K-method. Cwikel and Peetre showed that if a given Banach couples A is a K-monotone space, all interpolation spaces can be constructed by the Peetre K-method. But, they really show only that all interpolation cones can be constructed by the Peetre K-method, 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 $\lambda$-K-monotone 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 $\lambda$-K-monotone 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
Description:98 p.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1988.
Other Identifier(s):(UMI)AAI8908741
Date Available in IDEALS:2014-12-16
Date Deposited:1988

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