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?


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