The standard functional isoperimetric inequality is for an integer $n\ge 1$, $$ \Vert u\Vert_{L^{\frac{n}{n-1}}(\mathbb R^n)}\le c(n)\Vert \nabla u\Vert_{L^1(\mathbb R^n)}, \quad c(n)=\frac{(\vert\mathbb B^n_2\vert_n)^{\frac{n-1}{n}}}{ \vert\mathbb S^{n-1}_2\vert_{n-1} }, \tag{$\flat$}$$ where $\vert \cdot\vert_d$ is the $d$-dimensional Hausdorff measure, $\mathbb B^n_2$ is the unit standard Euclidean ball of $\mathbb R^n,$ $\mathbb S^{n-1}_2$ is the unit standard Euclidean sphere, $u$ is any function in $W^{1,1}(\mathbb R^n)$ (which happens to be included in $L^{\frac{n}{n-1}}(\mathbb R^n)$).
Claim. It seems that this inequality can be improved as follows: for any $u\in W^{1,1}(\mathbb R^n)$, we have $$ \Vert u\Vert_{L^{\frac{n}{n-1}, 1}(\mathbb R^n)}\le c(n)\Vert \nabla u\Vert_{L^1(\mathbb R^n)}, \tag{$\sharp$}$$ where $L^{\frac{n}{n-1}, 1}(\mathbb R^n)$ is the (smallest) Lorentz space based upon $L^{\frac{n}{n-1}}$ (thus with the largest norm) and that we have in fact $$ W^{1,1}(\mathbb R^n)\subset L^{\frac{n}{n-1},1}(\mathbb R^n). $$ Question. Is the above improvement proven somewhere and if it is the case, what is the standard reference for that result?
