Two-weight inequalities for the Hilbert transform of monotone functions
The authors characterize the weighted inequality \[ \left( \int_{-\infty }^{\infty }\left| \mathcal{H} f(x)\right| ^{q}\left| x\right| ^{\alpha }dx\right) ^{\frac{1}{q} }\leq C\left( \int_{-\infty }^{\infty }\left| f(x)\right| ^{p}\left| x\right| ^{\beta }dx\right) ^{\frac{1}{p}} \] restricted to the cone of odd or even monotone functions, where \(\mathcal{H}\) is the Hilbert transform and \(0<p\leq q<\infty\). Furthermore, the discrete analogue of the above result is also established. Several consequences of the above results are examined and discussed.
scientific article; zbMATH DE number 5998378