Weighted norm inequalities for the conjugate function and Hilbert transform

The principal problem considered is the determination of all non-negative functions $W(x)$ with period $2\pi$ such that \[ \int _{ - \pi }^\pi {|\tilde f(\theta ){|^p}W(\theta )\;d\theta \leq C} \;\int _{ - \pi }^\pi {|f(\theta ){|^p}W(\theta )\;d\theta } \] where $1 < p < \infty$, f has period $2\pi$, C is a constant independent of f, and $\tilde f$ is the conjugate function defined by \[ \tilde f(\theta ) = \lim \limits _{\varepsilon \to {0^ + }} \frac {1}{\pi }\int _{\varepsilon \leq |\phi | \leq \pi } {\frac {{f(\theta - \phi )\;d\phi }}{{2\tan \phi /2}}.} \] The main result is that $W(x)$ is such a function if and only if \[ \left [ {\frac {1}{{|I|}}\int _I {W(\theta )\;d\theta } } \right ]{\left [ {\frac {1}{{|I|}}\int _I {{{[W(\theta )]}^{ - 1/(p - 1)}}d\theta } } \right ]^{p - 1}} \leq K\] where I is any interval, $|I|$ denotes the length of I and K is a constant independent of I. Various related problems are also considered. These include weak type results, the nonperiodic case, the discrete case, an application to weighted mean convergence of Fourier series, and an estimate for one of the functions in the Fefferman and Stein decomposition of functions of bounded mean oscillation.