If $f\in H^2(\mathbb R^2)$, I want to show that
- $||f||_{L^\infty}\le c||f||_{H^2}$
- $||f||_{L^\infty}\le c||f||_{H^1} [1+\ln(1+||f||_{H^2})]$
For 1, I use $||f||_{L^\infty}\le \sup_{x\in \mathbb R^2} |f|\le\int_{\mathbb R^2} |\hat f(\xi)|d\xi = \int(1+|\xi|^2)^{-1}(1+|\xi|^2)|\hat f(\xi)|d\xi \le (\int (1+|\xi|^2)^{-2}d\xi)^2(\text {which is integrable in this case})||f||_{H^2}\to \text{is this correct?}$
But for 2, I am stucked. How can I get the "ln"? and how can I make it into a product of $H^1$ and $H^2$ norm?