Weighted bilinear Hardy inequalities

We characterize the weights w, w 1 , w 2 such that the weighted bilinear Hardy inequality ( ∫ a b ( ∫ a x f ) q ( ∫ a x g ) q w ( x ) d x ) 1 q ⩽ C ( ∫ a b f p 1 w 1 ) 1 p 1 ( ∫ a b g p 2 w 2 ) 1 p 2 holds for all nonnegative functions f and g, with a positive constant C independent of f and g, for all possible values of q, p 1 and p 2 with 1 < q , p 1 , p 2 < ∞ . We also characterize the good weights for the weighted bilinear n-dimensional Hardy inequality to hold.