The Weighted Hardy Inequality: New Proofs and the Case <i>p</i> = 1

An elementary proof is given of the weight characterisation for the Hardy inequality ( ∫ 0 ∞ ( ∫ 0 ∞ f ) q υ ( x ) d x ) 1 / q ⩽ C ( ∫ 0 ∞ f p u ) 1 / p for f ⩾ 0 , (1.1) in the case 0 < q < p, 1 < p < ∞. It is also shown that certain weighted inequalities with monotone kernels are equivalent to inequalities in which one of the weights is monotone. Using this, a characterisation of those weights for which (1.1) holds with 0 < q < I = p is given. Results for (1.1), considered as an inequality over monotone functions f are presented.