Lattice Properties and Interpolation Theory of the Spaces Lambda(psi,q) and M(psi)

Lee, Chongsung

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Description

Title

Lattice Properties and Interpolation Theory of the Spaces Lambda(psi,q) and M(psi)

Author(s)

Lee, Chongsung

Issue Date

1988

Doctoral Committee Chair(s)

Peck, N. Tenney,

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(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 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$\sb{\psi\sb1}$).

Graduation Semester

1988

Type of Resource

text

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