Given $1\leq p<\infty$, the homogeneous Sobolev space $\dot{W}^{1,p} (\mathbb{R}^{N})$ is defined as the space of all functions $u\in L_{\operatorname*{loc}}^{1}(\mathbb{R}^{N})$ such that the distributional gradient $\nabla u$ belongs to $L^{p}(\mathbb{R}^{N};\mathbb{R}^{N})$. This space is a seminormed space and it can be made into a normed space by considering the quotient space $\dot{W}^{1,p}(\mathbb{R}^{N})/\mathbb{R}$.
In a classical paper "Some problems of vector analysis and generalized formulations of boundary-value problems for the Navier-Stokes equations" Ladyzhenskaya and Solonnikov consider the space $\dot{H}^{1,p}% (\mathbb{R}^{N})$ defined as the closure in $\dot{W}^{1,p} (\mathbb{R}^{N})$ of the space of functions $u\in C_{c}^{\infty}(\mathbb{R}^{N})$ such that $u_{B}=0$ with respect to the seminorm $\Vert\nabla u\Vert_{L^{p}(\mathbb{R}^{N})}$, that is, $$ \dot{H}^{1,p}(\mathbb{R}^{N}):=\overline{\{u\in C_{c}^{\infty}(\mathbb{R}% ^{N}):\,u_{B}=0\}}^{\dot{W}^{1,p}(\mathbb{R}^{N})}. $$ Here,$$ u_{B}:=\frac{1}{\mathcal{L}^{N}(B)}\int_{B}u\,dx. $$ The importance of this subspace is that the following Hardy-type inequalities hold in $\dot{H}^{1,p}(\mathbb{R}^{N})$, $N\geq2$, $$ \int_{\mathbb{R}^{N}}\frac{|u(x)-u_{B}|^{p}}{(1+\Vert x\Vert^{2})^{p/2}}dx\leq c\int_{\mathbb{R}^{N}}\Vert\nabla u(x)\Vert^{p}dx $$ for $p\neq N$ and for some constant $c=c(N,p)>0$, and $$ \int_{\mathbb{R}^{N}}\frac{|u(x)-u_{B}|^{N}}{(1+\Vert x\Vert^{2}\log^{2}\Vert x\Vert)^{N/2}}dx\leq c\int_{\mathbb{R}^{N}}\Vert\nabla u(x)\Vert^{N}dx $$ for $p=N$ and for some constant $c=c(N)>0$. A sketch of the proof for $p=N$ is given in the above paper.
I would like to find an example (if it exists) of a function which belongs to $\dot{W}^{1,N}(\mathbb{R}^{N})$ but does not satisfy the above Hardy's inequality (and so does not belong to $\dot{H}^{1,N}(\mathbb{R}^{N})$).
The standard pathological function in $\dot{W}^{1,N}(\mathbb{R}^{N})$ is $u(x)=\log\log\Vert x\Vert$ for $\Vert x\Vert\geq2$ and anything nice in $B(0,2)$. Unfortunately the left-hand side of the previous inequality is finite since $$ \int_{2}^{\infty}r^{N-1}\frac{|\log\log r-u_{B}|^{N}}{(1+r^{2}\log^{2}% r)^{N/2}}dr\leq c\int_{2}^{\infty}r^{N-1}\frac{|\log\log r|^{N}}{r^{N}\log ^{N}r}dr\leq c\int_{2}^{\infty}\frac{1}{r\log^{N-\varepsilon}r}dr<\infty. $$ I cannot think of anything else.
