Date of Award

3-1987

Degree Type

Thesis

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Supervisor

H.P. Heinig

Abstract

This thesis is concerned with the study of integral operators of the form (UNFORMATTED TABLE OR EQUATION FOLLOWS)$${\rm (Kf)(x)}=\int\sbsp{-\infty}{\infty}{\rm k(x,t)f(t)\ d}\mu{\rm (t)},\ {\rm x}\in{\rm I}\!{\rm R},$$(TABLE/EQUATION ENDS)between Lebesgue and "weak" Lebesgue spaces with general measures. For large classes of kernels we characterize the measures $\mu$ and $\nu$ for which the operator ${\rm K:L}\sbsp{\mu}{\rm p}\to{\rm L}\sbsp{\nu}{\rm q},$ or ${\rm K:L}\sbsp{\mu}{\rm p}\to$ weak ${\rm L}\sbsp{\nu}{\rm q},$ 0 $<$ q $<$ $\infty,$ 1 $\le$ p $<$ $\infty$ is bounded. If K is the Hardy operator our results are applied to prove a weighted Marcinkiewicz interpolation theorem and if K is the Stieltjes transform our characterization has an application to the Hilbert double series. In the case that K is the Fourier transform, we consider also the higher dimensional analogue and prove several weighted norm inequalities for near optimal weights. The one-dimensional form is applied to prove Laplace representation theorems for functions in weighted Bergman spaces.


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