Agmon's Inequality in higher dimensions

Question

We have the Agmon inequality on $[0,a]$ $$\| u \|_{L^{\infty}} \leq \|u \|_{L^2}^{1/2} \|u_x\|_{L^2}^{1/2}$$ Is there a version in two dimensions, say on $[0,a]^2$? I know there is a multidimensional form involving $H^s$-norms, but I would really like to keep the right hand side as a product of powers of $L^2$-norms of $u$ and its $x$-derivatives (preferably not $y$-derivatives).

Answer 1

No, in two dimensions the $L^2$ norm of the first derivative does not control the supremum of the function. A standard example is $$u(x,y)=\log\log (x^2+y^2)$$ (multiplied by a smooth cutoff function) which is unbounded, yet has square integrable gradient: $$ |\nabla u(x,y)| = \frac{1}{r \log r},\quad r=\sqrt{x^2+y^2} $$ The above is concisely expressed by saying that $H^{1}$ does not embed into $L^\infty$ in two dimensions.