Let $k,n \in \mathbb{N}$ such that $k>\frac{n}{2}$. How can one prove that $u\in W^{k,2}(\mathbb{R}^n)$ may be embedded into $L^\infty(\mathbb{R}^n)$?
$W^{k,2}(\mathbb{R}^n) \hookrightarrow L^\infty(\mathbb{R}^n)$
2
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functional-analysis
pde
sobolev-spaces
1 Answers
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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).