Norms in Weighted //-Spaces and Hardy Operators

Let w denote a weight function on (0, oo), i. e., a nonnegative measurable function on (0, oo). For 1 < p < oo the weighted space L p(w) is the space of real functions generated by the norm

with the usual modification for p = oo. The weighted Hardy operator Hw is defined by

when 0 < W(x) := w(t)dt < oo for all x > 0 (cf. [8]). Note that for w = 1 the operator Hw is the usual Hardy operator H f ( x ) = ( \ / x ) f ( t ) d t .