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I need to show $|f|_{L^\infty}\leq c|f|_{H^2} = c(\int_{\mathbb R^n} (1+|\xi|^2)^2|\hat f(\xi)|^2 d\xi )^{1/2}$, assume $f\in H^2(\mathbb R^2)$

I think I can trasnfer $f\ = \int \hat f(\xi)e^{2\pi i \xi\cdot x}d\xi$, and $|f(x)|=\int|\hat f(\xi)|d\xi$ and use Cauchy-Schwarz to get the $H^2$ norm. But I encounter a troubles:

  1. the infinity norm is the esssup $|f|$, but this representation kind of far away from this inequality. Is there any other way to represent the infinity norm?
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    I think this is the most direct proof. Since functions in $H^2(\mathbb{R}^2)$ are automatically continuous, the essential supremum is equal to the supremum.2012-12-15

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Take a look in the page 272, Proposition 1.5 from this book

In the demonstration, the author shows that $H^{1+\alpha}(\mathbb{R}^2)$ is continuously immersed in $C^\alpha(\mathbb{R}^2)$ for $\alpha\in (0,1)$. On the other hand, $H^2(\mathbb{R}^2)$ is continuously immersed in $H^{1+\alpha}(\mathbb{R}^2)$ and $C^\alpha(\mathbb{R}^2)$ is continuously immersed in $C^0(\mathbb{R}^2)$, hence you can conclude.

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    @TianyiXia, Im using a abusive notation but here $C^{\alpha}(\mathbb{R}^2)$ and $C^0(\mathbb{R}^2)$ mean the space of functions $f$ such that \|f\|_\alpha<\infty and \|f\|_0<\infty respectivley, where \|f\|_\alpha<\infty is the Holder norm and \|f\|_0<\infty is the sup norm.2012-12-18