Suppose $\omega$ is defined on torus $\mathbb{T}$ with mean-zero condition, $\langle\omega\rangle=\int_{\mathbb{T}}\omega(\alpha)\mathrm{d}\alpha=0.$
Suppose $h(\alpha)=H\omega(\alpha)=P.V.\frac{1}{\pi}\int_{\mathbb{T}}\frac{\omega(\beta)}{\alpha-\beta}\mathrm{d}\beta$ is well-defined on for $\alpha\in\mathbb{T}$.
I was wondering if the Hilbert transform can commute with integration, i.e., $\int_{\mathbb{T}}h(\alpha)\mathrm{d}\alpha=\int_{\mathbb{T}}H\omega(\alpha)\mathrm{d}\alpha=H\int_{\mathbb{T}}\omega(\alpha)\mathrm{d}\alpha=0$.
I am not familiar with singular integral operators, and I do not know how to apply Fubini's theorem here.