On weighted norm inequalities for the Hilbert transform of functions with moments zero

Abstract:

Let $\tilde f$ denote the Hilbert transform of $f$, i.e. \[ \tilde f(x) = {\rm {p}}{\rm {.v}}{\rm {.}}\int {\frac {{f(t)}}{{x - t}}dt} \] and let $1 < p < \infty$. A weight function $w$ is shown to satisfy \[ \int {|\tilde f(x)} {|^p}w(x)dx \le C{\int {|f(x)|} ^p}w(x)dx\] for all $f$ with the first $N$ moments zero, if and only if it is of the form $w(x) = |q(x){|^p}U(x)$, where $q$ is a polynomial of degree at most $N$ and $U \in {A_p}$.