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

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