# AN INTEGRAL INEQUALITY WITH APPLICATIONS

#### Abstract

The integral inequality in this thesis has been of interest ever since a simple version of it was proven in 1971 by J. Moser. He established limits of exponential integrability of a certain space of Sobolev functions. In the context of convex functions this thesis generalizes and provides necessary and sufficient conditions for the inequality to exist in general, and applies the results to obtain exponential integrability type inequalities for some spaces of Sobolev functions.^ Let 1 (LESSTHEQ) q (.) < (INFIN), 1/p + 1/q = 1, f (ELEM) L('p)((//R)), and g (ELEM) L('q)((//R)) with (PARLL)g(PARLL)(,q) (LESSTHEQ) 1. Let (psi)(r) = (PARLL)f(.)(chi)(,{0,r})(PARLL)p and F(r) = (PARLL)f(.)g(.)(chi)(,{0,r})(PARLL)(,1). Then we have (psi)(r) (GREATERTHEQ) F(r) by Holder's inequality. If (PHI) is a positive nonincreasing function on {0,(INFIN)) and N is a convex function, we investigate when the integral of (PHI){N{(psi)(r)}-N{F(r)}} with respect to the measure induced by N{(psi)(r)} is bounded by (PARLL)(PHI)(PARLL)(,1). This provides an indication of how bad Holder's inequality is most of the time. Chapter 1 establishes the inequality for a specific class of convex functions.^ In Chapter II applications are given by establishing integral inequalities of certain spaces of Sobolev functions. ^

#### Degree

Ph.D.

#### Subject Area

Mathematics

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