## New estimate for elliptic system

$\mathbf{Problem:}$ Suppose $\boldsymbol{f}=(f_1,f_2,\cdots,f_n)\in C^1(\mathbb{R}^n)$. Assume $\nabla\cdot\boldsymbol{f}=0$

If $\boldsymbol{u}$ satisfies $\displaystyle \Delta \boldsymbol{u}=\boldsymbol{f}$, consider its classical solution $\boldsymbol{u}=\Gamma\ast \boldsymbol{f}$

Then                     $||\nabla\boldsymbol{u}||_{L^\frac{n}{n-1}(\mathbb{R}^n)}\leq C||\boldsymbol{f}||_{L^1(\mathbb{R}^n)}$

and hence                   $||\boldsymbol{u}||_{L^\frac{n}{n-2}(\mathbb{R}^n)}\leq C||\boldsymbol{f}||_{L^1(\mathbb{R}^n)}$

$\mathbf{Proof:}$

$\text{Q.E.D}\hfill \square$

$\mathbf{Remark:}$ Jean Bourgain · Ha¨ım Brezis.New estimates for elliptic equations and Hodge type systems