Estimate for functions in Sobolev space $H^s$

Question

Can anyone help me with this question

Use the Fourier transform to prove that if $u\in H^s(\mathbb{R}^n)$ for an integer $s$ such that $s>n/2$ then $u\in L^\infty (\mathbb{R}^n)$.

Answer 1

Work first with Schwartz functions. We calculate $$ |u(x)| \leq \| \check{u} \|_{L^1} = \int_{\mathbb{R}^n} (1+|\xi|^2)^{-s/2}(1+|\xi|^2)^{s/2} |\check{u}| d \xi. $$

Now just notice that if $s>n/2$ then $(1+|\xi|^2)^{s/2}\in L^2(\mathbb{R}^n)$ so you can apply Hölder's inequality and conclude that

$$ \| u\|_{L^\infty} \leq C(n,s) \| u\|_{H^s}. $$ Now use a limiting argument.