积分不等式

Assume $f,g,\phi\in C_c^\infty(\bbR^3)$, and $\sed{i,j,k}$ is a permutation of $\sed{1,2,3}$. Then there exists a generic constant $C$ such that      $$\bex      \iiint_{\bbR^3}|fg\phi|\rd x      \leq C\sen{f}_{L^2}      \sen{g}_{L^2}^\frac{1}{4}      \sen{\p_ig}_{L^2}^\frac{1}{2}      \sen{\p_kg}_{L^2}^\frac{1}{4}      \sen{\phi}_{L^2}^\frac{1}{4}      \sen{\p_j\phi}_{L^2}^\frac{1}{2}      \sen{\p_k\phi}_{L^2}^\frac{1}{4}.      \eex$$      

 

Proof.      Without loss of generality, we may assume $i=1,j=2,k=3$.      $$\beex      \bea      \iiint |fg\phi|\rd x      &\leq \iint \sup_{x_1}|g|\cdot \sen{\phi}_{L^2_{x_1}}\cdot \sen{f}_{L^2_{x_1}}\rd x_2\rd x_3\\      &\leq C\iint      \sen{g}_{L^2_{x_1}}^{1/2}      \sen{\p_1 g}_{L^2_{x_1}}^{1/2}      \cdot \sen{\phi}_{L^2_{x_1}}\cdot \sen{f}_{L^2_{x_1}}\rd x_2\rd x_3\\      &\leq C\int      \sen{g}_{L^2_{x_1x_2}}^{1/2}      \sen{\p_1g}_{L^2_{x_1x_2}}^{1/2}\cdot      \sen{f}_{L^2_{x_1x_2}}\cdot \sup_{x_2}\sen{\phi}_{L^2_{x_1}}\rd x_3\\      &\leq C\int      \sen{g}_{L^2_{x_1x_2}}^{1/2}      \sen{\p_1g}_{L^2_{x_1x_2}}^{1/2}\cdot      \sen{f}_{L^2_{x_1x_2}}      \cdot \sen{\phi}_{L^2_{x_1x_2}}^{1/2}      \sen{\p_2\phi}_{L^2_{x_1x_2}}^{1/2}\rd x_3\\      &\leq C\sen{\p_1g}_{L^2}^{1/2}      \sen{\p_2\phi}_{L^2}^{1/2}      \sen{f}_{L^2}      \sup_{x_3}\sex{      \sen{g}_{L^2_{x_1x_2}}^{1/2}      \sen{\phi}_{L^2_{x_1x_2}}^{1/2}}\\      &\leq  C\sen{\p_1g}_{L^2}^{1/2}      \sen{\p_2\phi}_{L^2}^{1/2}      \sen{f}_{L^2} \sen{g}_{L^2}^{1/4}      \sen{\p_3g}_{L^2}^{1/4}      \sen{\phi}_{L^2}^{1/4}      \sen{\p_3\phi}_{L^2}^{1/4}.      \eea      \eeex$$      

 

This follows from [F. Wang, K.Y. Wang,      Global existence of $3$D MHD equations with mixed partial dissipation and magnetic diffusion.      Nonlinear Analysis: Real World Appl., 14 (2013), 526--535].