Hardy operators

Abstract

The maps T we shall consider in this chapter act between Lebesgue spaces on an interval (a,b), where b may be infinite, and are of the form $$Tf(x)=v(x)\int_{a}^{x}u(t)f(t)dt,$$ u and v being prescribed functions. They are commonly called Hardy operators, or operators of Hardy type, the operator originally studied by Hardy being that in which a=0,b= 8 and v=u= 1. Necessary and sufficient conditions for the boundedness or compactness of T are given.When u and v are both identically equal to 1 and b is finite, the exact value of the norm of T is determined; it is shown that it is attained at a function expressible in terms of generalised trigonometric functions.