Linear truncations of the Hilbert transform (Q1374611)

The author considers the operator \(H_{a,b,c,d}f(x)=\int_{ax+b}^{cx+d} dt/t\) arising in connection with questions about bilinear operators of type \(\int_{R^1}g(t-ax)f(x-t) dt/t\), generalizing the Hilbert transform, but for \(a\neq 0\) being no convolutions. The following problems are studied here: 1) Boundedness of \(H_{a,b,c,d}:L^p\to L^q\) and an estimate for its norm, \(a\neq 0\), \(1<p\leq \infty \), \(1\leq q\leq p\), \(q<\infty\). 2) For the same range of parameters, an analog of the previous item for \(I_{a,b,a,d}\), where \(I_{a,b,c,d}\) is obtained from \(H_{a,b,c,d}\) by considering the bounds for the integral equal to \(a|x|+b\) and \(c|x|+d\). 3) Boundedness of \(I_{a,b,c,d}\) in \(L^p\) for \(1<p<\infty \), \(a\neq 0\), \(c\neq 0\).