An elementary proof is given of the weight characterisation for the Hardy inequalityIt 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 / are presented.

Abstract. Let p ∈ (0, ∞) , let v be a weight on (0, ∞) and let Λ p (v) be the classical Lorentz space, determined by the norm

Necessary and sufficient conditions for the boundedness from L%(U + ) to L*((R + ) of Volterra integral operators of the form K/(x)= \'k(x,y)f(y)dy, Jo where k(x,y) is a non-negative kernel under suitable monotone conditions, are given. The cases 1 < / ? < < ? < oo, 1 < q < p < oo and 0 < g < l < / ? < o o a r e considered. The results extend the well-known weighted norm Hardy inequality. k(x,y) = ^r ) (x-yy-1 f o r r^l when 1