## Digitized Theses

1995

Dissertation

#### Degree Name

Doctor of Philosophy

#### Abstract

This thesis is devoted to the study of integral operators of the form{dollar}{dollar}Kf(x)=\int\sbsp{lcub}0{rcub}{lcub}+\infty{rcub} k(x,t)f(t)dt{dollar}{dollar}on weighted Orlicz spaces. Weight characterizations are obtained for weighted modular inequalities, which generalize the results by Q. Lai and by H. Heinig and L. Maligranda for the Hardy operator. We also give results that parallel those by S. Bloom and R. Kerman for operators with more general kernels, but our results are valid under weaker conditions. Our results also have applications to the Stieltjes transformations and Hardy's inequalities for higher order derivatives.;Furthermore, the results above can be used to characterize the weights for modular inequalities when K is restricted to the monotone functions. These results generalize the corresponding ones in Lebesgue spaces proved by H. Weinig and A. Kufner for the Hardy operators and by V. Stepanov for a class of Volterra convolution operators.;Also, for the identity operator, which is a special case of an integral operator, we study both the modular and norm inequalities.;Finally, a reverse Holder inequality in Orlicz spaces is studied and the result is new even in the Lebesgue space case.

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