Hilber transform on [0,1)

Question

Let $\mathbb{T}=[0,1)$ and $H$ be a Hilbert transform on $L^p(\mathbb{T})$ when $2\leq p< \infty$. If $f$ is $L^p$ and $f_n$ is trignometric polynomial such that $f_n\rightarrow f$ in $L^p$ sense. How do you prove that $f_nHf\rightarrow fHf$ in $L^{p/2}(\mathbb{T})$ sense. Thanks

Answer 1

Use Cauchy-Schwarz's inequality: $$ \int_{\mathbb{T}}|f_n\,Hf-f\,Hf|^{p/2}=\int_{\mathbb{T}}|f_n-f|^{p/2}\,|Hf|^{p/2}\le\Bigl(\int_{\mathbb{T}}|f_n-f|^{p}\Bigr)^{1/2}\Bigl(\int_{\mathbb{T}}|Hf|^{p}\Bigr)^{1/2}. $$