Generalization of Gagliardo-Nirenberg Inequality The standard Gagliardo-Nirenberg Inequality is
$$
\Vert u\Vert_{L^{\frac{n}{n-1}}(\mathbb R^n)}\le C_n \Vert \nabla u\Vert_{L^{1}(\mathbb R^n)},
\tag{$\ast$}$$
and constitutes a key step to proving Sobolev Injection Theorems. Of course we may assume that $u$ is smooth and compactly supported, the important point is that the constant $C_n$ is dimensional and independent of $u$.
Question 1. Are there  some improvements of ($\ast$) where the rhs is replaced by a quasi-norm of $\nabla u$ in weak $L^1$?
Question 2. Are there  some improvements of ($\ast$) where the lhs is replaced by a norm of $u$ in the Lorentz space $L^{\frac{n}{n-1}, 1}$ or $L^{\frac{n}{n-1}, p}$
with $p<n/(n-1)$?
Question 3. Probably linked to the previous questions: is the following inequality true ? For $n\ge 3$,
$$
\Vert u\Vert_{L^{\frac{n}{n-2}}(\mathbb R^n)}\le C_n \Vert \Delta u\Vert_{L^{1}(\mathbb R^n)}.
$$
 A: Question 1 I think the short answer is no. Try testing functions of the form $|x|^{-n + 1}$ around $0$. Its gradient would be like $|x|^{-n}$ and thus in $L^{1,\infty}$ but the function is not in the required $L_p$ space. The Sobolev inequalities for $p=1$ are connected with the isoperimetric inequality, see [S] and are thus very tight. I recall a result similar to what you want but at the other end of the range, i.e. if $\nabla u \in L^{n,\infty}$ then $u$ is in BMO. 
Question 2 Try expressing $v = \nabla u$ as $u = D_n \ast v$ for some $\mathbf{R}^n$-valued distribution $D_n$. Then check in which Lorentz spaces $D_n$ is and apply the Young's inequalities in Lorentz spaces, as in [N].
Question 3 This would be immediately correct with weak $n/n-2$ as a consequence of Hardy-Littlewood-Sobolev inequalities, which hold for every fractional power of $(-\Delta)$ i.e.:
$$
  \| \, u \,  \|_{L^{\frac{n}{n - \alpha},\infty}} \lesssim \| \, (-\Delta)^{\frac{\alpha}{2}} u \, \|_{L^1}
$$
see [V]. When $\alpha = 2$, the inequality would still be true without weak bounds after changing $\Delta$ by the double gradient $\nabla^2$ by iteration of the Gagliardo-Niremberg inequality. I do not think that you can change the $\nabla^2$ by a $\Delta$ unless the iterated Riesz transforms $R_i R_j$ were bounded in $L^1$, which they are not. To produce an explicit counterexample i would first show that the inequality if tight for $\nabla^2$ ans then use that $\|\nabla^2 u\|_1$ and $\|\Delta u\|_1$ are not comparable.

[N] Nursultanov, Erlan; Tikhonov, Sergey, Convolution inequalities in Lorentz spaces, J. Fourier Anal. Appl. 17, No. 3, 486-505 (2011). ZBL1235.44012.https://link.springer.com/article/10.1007/s00041-010-9159-9
[S] Saloff-Coste, Laurent, Aspects of Sobolev-type inequalities, London Mathematical Society Lecture Note Series. 289. Cambridge: Cambridge University Press. x, 190 p. (2002). ZBL0991.35002.
[V] Varopoulos, N. Th.; Saloff-Coste, L.; Coulhon, T., Analysis and geometry on groups, Cambridge Tracts in Mathematics 100. Cambridge: Cambridge University Press (ISBN 978-0-521-08801-5/pbk). xii, 156 p. (2008). ZBL1179.22009.> Blockquote
