(Multiplicative Sobolev inequality). Let $mu,lambda$ and $gamma$ be three parameters that satisfy $$ex 1leq mu,lm<infty,quad frac{2}{lm}+frac{1}{mu}>1quadmbox{and}quad 1+frac{3}{gamma}=frac{2}{lm}+frac{1}{mu}. eex$$ Assume $phiin H^1(bR^3)$, $p_1phi$, $p_2phiin L^lm(bR^3)$, $p_3phiin L^mu(bR^3)$. Then, there exists a constant $C=C(mu,lm)$ such that $$ex sen{phi}_gammaleq Csen{p_1phi}_{L^lm}^frac{1}{3} sen{p_2phi}_{L^lm}^frac{1}{3} sen{p_3phi}_{L^mu}^{frac{1}{3}}. eex$$
Reference:
C.S. Cao, J.H. Wu, Two regularity criteria for the $3$D MHD equations, J. Differential Equations, 248 (2010), 2263--2274.