If $\mathcal{H}$ denotes Hilbert transform and $\theta\in H^{2}$ classical Sobolev space. How to prove that there is $C>0$ such that $$||\theta_x||_{L^{\infty}}+||\mathcal{H}\theta_x||_{L^{\infty}}\leq C ||\theta||_{H^{2}}.$$
2026-03-25 03:04:10.1774407850
Inequality $||\theta_x||_{L^{\infty}}+||\mathcal{H}\theta_x||_{L^{\infty}}\leq C ||\theta||_{H^{2}}$
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