Let $k,n \in \mathbb{N}$ such that $n>\frac{k}{2}$. How can one prove that $u\in W^{k,2}(\mathbb{R}^n)$ may be embedded into $L^\infty(\mathbb{R}^n)$?
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We can use Fourier analysis. First, if $u$ is a test function, we can write $$u(x)=C\int_{\Bbb R^n}\widehat u(t)e^{itx}(1+|t|^2)^{k/2}(1+|t|^2)^{-k/2}dt.$$ Then using Cauchy-Schwarz inequality, we have $$|u(x)|^2\leqslant C^2\lVert u\rVert_{H^k}^2\int_{\Bbb R}(1+|t|^2)^{-k}dt=C^2\lVert u\rVert_{H^k}^2\omega_n^2\int_{\Bbb R}(1+s^2)^{-k}s^{n-1}ds.$$ Then we conclude by density of test functions (using a sequence converging also almost everywhere). |
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