Weighted Hardy inequalities and Hardy transforms of weights

Many problems in analysis are described as weighted norm inequalities that have given rise to different classes of weights, such as A(p)-weights of Muckenhoupt and B-p-weights of Arino and Muckenhoupt. Our purpose is to show that different classes of weights are related by means of composition with classical transforms. A typical example is the family M-p of weights omega for which the Hardy transform is L-p(omega)-bounded. A B-p-weight is precisely one for which its Hardy transform is in M-p, and also a weight whose indefinite integral is in A(p+1).