## Open Access Dissertations and Theses

3-1987

Thesis

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

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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