Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $L^\infty(\mathbb{R}^n)$?

share|cite|improve this question

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).

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.