# Sobolev space inequality

If $f\in H^2(\mathbb R^2)$, I want to show that

$||f||_{L^\infty}\le c||f||_{H^1} [1+\ln(1+||f||_{H^2})]$

How can I get the "ln"? and how can I make it into a product of $H^1$ and $H^2$ norm?

It is actually one of my Real variables's project. So how can I do this inequality in an relativly "elementary" way?

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I have not tried this myself, but you could try to use the Fourier transform. – Christopher A. Wong Dec 21 '12 at 10:14
This is an exact duplicate of no.2 in math.stackexchange.com/questions/259336/sobolev-inequality where an answer was given. The answer may not have been exactly what you were looking for, but this is no reason to create a duplicate. – user53153 Dec 22 '12 at 2:19