The Hilbert transform of $f$ is $$Hf(t)=\lim_{y\to 0^+}[Q_y(t)*f(t)]$$ where $Q_y(t)=\frac{t}{\pi(t^2+y^2)}$ is the conjugate Poisson kernel. Since $$\lim_{y\to 0^+}Q_y(t)=P.V\frac{1}{\pi t}$$ So the Hilbert transform of $f$ is given by $$Hf(t)=P.V\frac{1}{\pi t}*f(t)$$
Im a little confused by the statement. It seems that the order of limit and convolution are changed? Is it correct to change the order? Or I misunderstood ?