# Hilbert transform on a Besov space

Consider the usual Hilbert transform of periodic functions $$H(f) = \frac{1}{2\pi}P.V.\int_{-\pi}^{\pi}\cot(\frac{x-y}{2})f(y)dy.$$ We know $$H$$ does not map $$L^\infty$$ continuously to $$L^\infty$$. Now we consider $$H$$ on the Besov space $$B^0_{\infty, \infty}$$, which is a little larger than $$L^\infty$$. Can we say $$H$$ maps $$B^0_{\infty, \infty}$$ continuously to $$B^0_{\infty, \infty}$$? If no, are there any counter-examples?