# Logarithmic Sobolev embeddings

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?

• This works, you can just estimate $\|\widehat{u}\|_1$ with Cauchy-Schwarz. (I assume that $|D|$ is multiplication by $|\xi|$ on the Fourier transform side.) Commented May 23, 2021 at 20:43
• You may look at the Brezis-Gallouet inequality. Commented May 23, 2021 at 22:44
• @RaffaeleScandone, it might be helpful to state the inequality and provide a reference. Commented May 23, 2021 at 22:45

$$$$\|f\|_{L^{\infty}(\mathbb{R}^d)}\lesssim 1+\|f\|_{W^{d/p,p}(\mathbb{R}^d)}\ln^{(p-1)/p}(e+\|f\|_{W^{d/q+\alpha,q}(\mathbb{R}^d)}),\qquad p,q\in(1,\infty),\;\alpha>0.$$$$
Following Christian Remling suggestion, it seems that $$\Vert u\Vert_{L^\infty(\mathbb R^d)}\le \gamma(d)\bigl\{ \Vert u\Vert_{L^2(\mathbb R^d)}+ \Vert \vert D\vert^{d/2} L(\vert D\vert) u\Vert_{L^2(\mathbb R^d)} \bigr\}$$ with $$\int_1^\infty\frac{dr}{L(r)^2 r}<+\infty.$$