ABSTRACT
An example of such a K is the Riemann-Liouville kernel Ky (x, y) = (x — y )^, y > 0; another is
We are interested here in weighted modular inequalities of the form
THEOREM 1.1. Let <Hv) = f * <p(y)dy bean TV -function and let 'I 'O ) = <p~l (y)dy be its complementary N -function. Then, an operator T = T/c of Hardy type satisfies (1.2) if and only if
and
Now (1.3) is necessary and sufficient for the weak-type inequality
while (1.4) is equivalent to the following weak-type inequality involving the dual operator (T ’g) (a ) = / t°° K (y, x)g(y) d y ,
In Section 3 we obtain other characterizations of the weights in (1.5) and (1.6) that will be needed later on.
