Let $\mathscr H\,f$ denote the Hilbert transform of a function $f \in L^2(\mathbb R)$. We know that $\mathscr H$ is an isometry on $L^2(\mathbb R)$, but I want to know to what is the mapping properties of the Hilbert transform when acting on weighted spaces $L^2_{\delta}(\mathbb R)$ with $\delta>0$, where $$ \|f\|_{L^2_\delta(\mathbb R)}^2=\int_{\mathbb R} |f(x)|^2 (1+|x|^2)^\delta\,dx.$$
Also as a follow up question, let us consider the equation $\frac{\partial}{\partial \bar{z}} u=0$ on the upper half plane, subject to $\Re u=f$ on the real-axis. What can we say about the solution $u$ if we assume that $f \in L^2_\delta(\mathbb R)$.
Thanks,
