Date of Award

1989

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Lutz Weis

Abstract

The goal of this thesis was to isolate classes of bounded linear operators in $L\sb{p}(I)$ which on the one hand still have some of the well-known and useful properties of positive operators, but which on the other hand are large enough to include some important classes of operators (e.g. the Hilbert transform and the singular operators derived from it) that cannot be dominated by positive operators. In Chapter I, we study as a first class of this kind the $L\sb{p}$ regular operators. By definition such operators map equiintegrable sets in $L\sb{p}(I)$ into equiintegrable sets in $L\sb{p}(I)$ and sets compact in measure into sets compact in measure. We show that with respect to duality and pertubation theory they have properties similar to positive operators. In Chapter II, we study strongly $L\sb{p}$ regular operators as the class of operators, which preserves growth restrictions of $L\sb{p}$ functions (formulated in terms of nonincreasing rearrangements of functions). We show that such operators can be extended to bounded linear operators on certain Lorentz and Marcinkiewicz spaces. Many important operators in analysis are in this class since we can show that all interpolated operators are strongly $L\sb{p}$ regular. Chapter III contains some representation theorems for linear operators in $L\sb{p}(I)$ by kernels of distributions, which are motivated by the representation of positive operators by stochastic kernels.

Pages

95

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