Weighted \(L^p-L^q\) inequalities for two-dimensional Hardy operators when \(q<p\) (Q1580297)

Let \(H\) be a two dimensional Hardy operator defined by \[ (Hf)(x,y) = \int^x_0 \int^y_0 f(x,y) dx dy, \quad 0 < x,y < \infty, \] and let \(u\) and \(v\) be weight functions on \((0,\infty)\times(0,\infty)\). The aim of the paper is to establish sufficient conditions for the boundedness of \(H\) from the weighted Lebesgue space \(L^p((0,\infty)\times (0,\infty);v)\) into the weighted Lebesgue space \(L^q((0,\infty)\times (0,\infty);u)\) provided that \(0<q<\infty\), \(1<p<\infty\) and \(q<p\). (Note that in the case when \(p\leq q\) the boundedness was characterized by \textit{E. Sawyer} [Stud. Math. 82, 1-16 (1985; Zbl 0585.42020)]).