I am trying to prove the Gagliardo--Nirenberg type inequality: $$ \Vert\nabla u\Vert_{L^{2p}(\mathbb{R}^{N})}\leq c|u|_{\operatorname*{BMO}% (\mathbb{R}^{N})}\Vert\nabla^{2}u\Vert_{L^{p}(\mathbb{R}^{N})}% $$ for every $u\in\operatorname*{BMO}(\mathbb{R}^{N})$ with $\nabla^{2}u\in L^{p}(\mathbb{R}^{N};\mathbb{R}^{N\times M})$ , where $N\geq2$ and $1<p<\infty$. This inequality has been proved for $u\in W^{2,p}(\mathbb{R}% ^{N})$ in the paper of P. Strzelecki "Gagliardo--Nirenberg inequalities with a BMO term", Bull. London Math. Soc. 38 (2006) 294--300. The file is here
https://www.mimuw.edu.pl/~pawelst/papers/Strzelecki-GNinequalitiesBMO-BullLMS2006.pdf The first step of the proof is to prove the inequality for $u\in C_{c}% ^{\infty}(\mathbb{R}^{N})$. Then one has to use density of $C_{c}^{\infty }(\mathbb{R}^{N})$ in $\operatorname*{BMO}(\mathbb{R}^{N})\cap W^{2,p}% (\mathbb{R}^{N})$. The idea is the following. Let's assume for the time being that $u\in C^{\infty}(\mathbb{R}^{N})\cap W^{2,p}(\mathbb{R}^{N})$. Let $Q_{n}:=(-n,n)^{N}$, let $\varphi\in C_{c}^{\infty}(\mathbb{R}^{N})$ be a smooth mollifier with support in $Q_{1}$ and let $\phi\in C_{c}^{\infty }(\mathbb{R}^{N})$ be such that $0\leq\phi\leq1$, $\phi=1$ in $Q_{1}$ and $\phi=0$ outside $Q_{2}$. Define $\phi_{n}(x):=\phi(x/n)$, $x\in\mathbb{R}% ^{N}$, and $$ u_{n}:=(u-u_{Q_{n}})\phi_{n}, $$ where $u_{Q_{n}}$ is the average of $u$ over $Q_{n}$. One can show that $|u-u_{n}|_{\operatorname*{BMO}(\mathbb{R}^{N})}\rightarrow0$ and that $\Vert\nabla^{2}u-\nabla^{2}u_{n}\Vert_{L^{p}}\rightarrow0$. Unfortunately to prove that $\Vert\nabla^{2}u-\nabla^{2}u_{n}\Vert_{L^{p}}\rightarrow0$ one need to use that $u\in W^{2,p}(\mathbb{R}^{N})$ and not just that $\nabla ^{2}u\in L^{p}(\mathbb{R}^{N};\mathbb{R}^{N\times M})$. Since $$ \nabla^{2}u_{n}=(u-u_{Q_{n}})\nabla^{2}\phi_{n}+2\nabla\phi_{n}\otimes\nabla u+\phi_{n}\nabla^{2}u, $$ one needs to use Poincare's inequality $$ \int_{Q_{n}}|u(x)-u_{Q_{n}}|^{p}dx\leq cn^{p}\int_{Q_{n}}|\nabla u|^{p}dx $$ in $W^{1,p}(\mathbb{R}^{N})$ to control the first term. You get $$ \int_{Q_{n}}|(u-u_{Q_{n}})\nabla^{2}\phi_{n}|^{p}dx\leq\frac{c}{n^{p}}% \int_{Q_{n}}|\nabla u|^{p}dx\rightarrow0 $$ provided $\nabla u\in L^{p}$. If I only know that $\nabla^{2}u\in L^{p}(\mathbb{R}^{N};\mathbb{R}^{N\times M})$ then I cannot conclude that $\nabla u\in L^{p}$.
On the other hand, there is a paper of P. Hajasz and A. Kalamajska, Polynomial asymptotics and approximation of Sobolev functions. Studia Math. 113 (1995), no. 1, 55-64.
http://matwbn.icm.edu.pl/ksiazki/sm/sm113/sm11314.pdf
In this paper they prove that if $\nabla^{2}u\in L^{p}(\mathbb{R}% ^{N};\mathbb{R}^{N\times M})$ with $1<p<\infty$ then there is a sequence of function in $C_{c}^{\infty}(\mathbb{R}^{N})$ such that $\Vert\nabla ^{2}u-\nabla^{2}v_{n}\Vert_{L^{p}}\rightarrow0$. The idea is similar to the one above and it is to take instead of cubes the annulus $A_{n}% :=B(0,2n)\setminus\overline{B(0,n)}$ and define $$ v_{n}:=(u-p_{n})\phi_{n}, $$ where now $p_{n}$ is a polynomial of degree one such that% $$ \int_{A_{n}}(u-p_{n})\,dx=0,\quad\int_{A_{n}}(\nabla u-\nabla p_{n})\,dx=0. $$ The advantage is that now one can use the Poincare inequality in $W^{2,p}$ instead of $W^{1,p}$ to get% \begin{align*} \int_{A_{n}}|u-p_{n}|^{p}dx & \leq cn^{2p}\int_{A_{n}}|\nabla^{2}u|^{p}dx,\\ \int_{A_{n}}|\nabla u-\nabla p_{n}|^{p}dx & \leq cn^{p}\int_{A_{n}}% |\nabla^{2}u|^{p}dx. \end{align*} Then $\nabla^{2}v_{n}=(u-p_{n})\nabla^{2}\phi_{n}+2\nabla\phi_{n}% \otimes(\nabla u-\nabla p_{n})+\phi_{n}\nabla^{2}u$ and so one can show that% $$ \int_{A_{n}}|(u-p_{n})\nabla^{2}\phi_{n}|^{p}dx\leq\frac{cn^{2p}}{n^{2p}}% \int_{A_{n}}|\nabla^{2}u|^{p}dx=c\int_{A_{n}}|\nabla^{2}u|^{p}dx\rightarrow0 $$ as $n\rightarrow\infty$ and a similar estimate holds for $\nabla\phi _{n}\otimes(\nabla u-\nabla p_{n})$. Note that $Q_{n}$ would not work here.
OK so my problem is that I have one good sequence $u_{n}$ to approximate the $BMO$ term but that does not work if I only assume that $\nabla^{2}u\in L^{p}(\mathbb{R}^{N};\mathbb{R}^{N\times M})$ and a different sequence $v_{n}$ to approximate the $\Vert\nabla^{2}u\Vert_{L^{p}(\mathbb{R}^{N})}$ term. So I have density of $C_{c}^{\infty}(\mathbb{R}^{N})$ in $BMO$ and in the homogeneous Sobolev space $\dot{W}^{2,p}(\mathbb{R}^{N})$, but I don't know how to produce a sequence that works for both spaces at the same time. Any suggestions? Thanks.
