乘积型Sobolev不等式

(Multiplicative Sobolev inequality). Let $\mu,\lambda$ and $\gamma$ be three parameters that satisfy $$\bex 1\leq \mu,\lm<\infty,\quad \frac{2}{\lm}+\frac{1}{\mu}>1\quad\mbox{and}\quad 1+\frac{3}{\gamma}=\frac{2}{\lm}+\frac{1}{\mu}. \eex$$ Assume $\phi\in H^1(\bbR^3)$, $\p_1\phi$, $\p_2\phi\in L^\lm(\bbR^3)$, $\p_3\phi\in L^\mu(\bbR^3)$. Then, there exists a constant $C=C(\mu,\lm)$ such that $$\bex \sen{\phi}_\gamma\leq C\sen{\p_1\phi}_{L^\lm}^\frac{1}{3} \sen{\p_2\phi}_{L^\lm}^\frac{1}{3} \sen{\p_3\phi}_{L^\mu}^{\frac{1}{3}}. \eex$$

Reference:

C.S. Cao, J.H. Wu, Two regularity criteria for the $3$D MHD equations, J. Differential Equations, 248 (2010), 2263--2274.