I am reading this paper about the Nonlinear Schrodinger Equation.
In Remark 2.9, the author mentions the following Sobolev inequality: For any compact time interval $I$, assume $0 \leq \sigma < \rho$, $1\leq r, r_1, q \leq \infty$. Then, $$ \|D^\rho u\|_{L_t^{q'}L_x^{r_1'}(I\times\mathbb{R}^d)} \lesssim \|D^\sigma u\|_{L_t^{q'}L_x^{r'}(I\times\mathbb{R}^d)}, $$
provided $r_1 = \frac{rd}{(\rho-\sigma)r+d}$, where $q', r_1'$ and $r'$ are the Hölder's conjugates of $q, r_1$ and $r$, respectively.
I assume that she is using some dual Sobolev inequality $$ \|D^\rho f\|_{L_x^{r_1'}(\mathbb{R}^d)} \lesssim \|D^\sigma u\|_{L_x^{r'}(\mathbb{R}^d)}, $$
valid for $\rho > \sigma$.
Question: Does this inequality really hold? Could you give some reference or some idea about proving this?
Take $\sigma=1$, $\rho=2$, and $d=2$. Then $r_1=\frac{2r}{r+2}>1$ provided $r>2$. So the inequality becomes (for functions depending only on space) $$\Vert D^2u\Vert_{L^{r_1'}(\mathbb{R}^2)}\le C \Vert Du\Vert_{L^{r'}(\mathbb{R}^2)}.$$ In particular, if you had a sequence of smooth functions with support contained in some ball $B$ and bounded in $W^{1,r'}(\mathbb{R}^2)$, then the sequence would be bounded in $W^{2,p}(B)$, where $p=\min\{r_1',r'\}$ and so by the Rellich-Kondrachov theorem up to a subsequence it would converge strongly in $W^{1,p}(B)$ and the gradients would converge pointwise a.e. This is false. Take $u_n(x)=\frac1n \sin(nx_1)\varphi(x)$, where $\varphi$ is a smooth function which is one in $[0,1]^2$ and zero outside of $[0,2]^2$. Then $Du_n$ is bounded uniformly in $L^\infty((0,2)^2)$ and so in any $L^{r'}((0,2)^2)$. But the gradients do not converge pointwise (not even up to a subsequence).