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 ine...
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