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 ${\displaystyle A_p}$-weights of Muckenhoupt and ${\displaystyle B_p}$-weights of Ariño 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 ${\displaystyle M_p}$ of weights w for which the Hardy transform is ${\displaystyle L_p\left(w\right)}$-bounded. A ${\displaystyle B_p}$-weight is precisely one for which its Hardy transform is in ${\displaystyle M_p}$, and also a weight whose indefinite integral is in ${\displaystyle A_{p+1}}$