It is well-known that $H^{\frac d2}(\mathbb R^d)=W^{\frac d2, 2}(\mathbb R^d)$ is not included in $L^\infty(\mathbb R^d)$, but it seems that there are some logarithmic substitutes. Is it true for instance that $$ \Vert u\Vert_{L^\infty(\mathbb R^d)}\lesssim \big\Vert \vert D \vert^{d/2}\ln(1+\vert D\vert) u\big\Vert_{L^2(\mathbb R^d)}\,, $$ or is there some analogous statement? Is there a clear connection with Gross' logarithmic Sobolev inequalities?