logarithmic sobolev inequalities I'm reading a paper about PDE in fluid dynamics, where it used something called logarithmic sobolev inequalities:
$$||\nabla u||_{\infty}\leq C||\omega||_{\infty} (1+\ln ||u||_m)$$
Where $\omega=\nabla \times u$ is the vorticity in $R^2$. How to prove this or anyone can give some reference about this topic?
 A: This is a classical result in singular integrals. Let me prove a similar inequality in the one dimensional case (for the several dimension case, you can read the paper by Kato and Ponce, Well-posedness of the Euler and NS equations in Lebesgue spaces, Revista Matemática Iberoamericana, 1986).
We want to prove the bound
$$
\|Hf\|_{L^\infty}\leq c\|f\|_{L^\infty}\max\left\{1,\log\left(\frac{\|f\|_{H^1}}{\|f\|_{L^\infty}}\right)\right\}+c\|f\|_{L^2},
$$
where $H$ is the Hilbert transform, i.e.,
$$
Hf(x)=\lim_{\epsilon\rightarrow0}\int_{|y|>\epsilon}\frac{f(y)}{x-y}dx.
$$
We split
$$
\int_{|y|>\epsilon}=\int_{\epsilon<|y|<\delta}+\int_{\delta<|y|<1}+\int_{1<|y|}=I_1+I_2+I_3.
$$
We have
$$
I_3\leq \|f\|_{L^2},
$$
$$
I_2\leq \|f\|_{L^\infty}\log\left(\frac{1}{\delta}\right).
$$
Finally, the integral $I_1$, can be written as
$$
I_1=\int_{\epsilon<|y|<\delta}\frac{f(x)-f(y)}{y-x}dx,
$$
thus,
$$
I_1\leq c\delta^{0.5}\|f\|_{C^{0.5}}\leq c\delta^{0.5}\|f\|_{H^1} \text{ (by Sobolev inequality)}.
$$
So, collecting everything, we have
$$
\|Hf\|_{L^\infty}\leq c(\delta^{0.5}\|f\|_{H^1}+ \|f\|_{L^\infty}\log\left(\frac{1}{\delta}\right)+\|f\|_{L^2}).
$$
We can take any $0<\delta\leq 1$. In particular, we can take
$$
\delta^{0.5}=\min\left\{1,\frac{\|f\|_{L^\infty}}{\|f\|_{H^1}}\right\}
$$
