Poincaré inequality and Rellich Theorem in one dimensional weighted Sobolev space

Question

Consider the weighted Sobolev space $W^{1,2}\big((0,R),r^{N-1}\big)$, $N=2,3,\ldots$ and its subspace $W_0^{1,2}\big((0,R),r^{N-1}\big)$. Anyone knows if the Poincaré inequality is true in this case? And if the $W^{1,2}\big((0,R),r^{N-1}\big)$ is compactly embedded in $L^2\big((0,R),r^{N-1}\big)$?

Answer 1

Let's write $p=N-1$.

Poincaré inequality

By density, it suffices to consider smooth functions $f\in W_0^{1,2}((0,R),r^{N-1})$. For $0<r<R$ we have $$r^{p} f(r)^2=r^{p} \left(\int_r^R f'(s)\,ds\right)^2 = r^{p} \left(\int_r^R s^{-p/2}f'(s)s^{p/2}\,ds\right)^2 \\ \overset{\rm Cauchy-Schwarz}{\le} r^p\int_r^R s^{-p}\,ds\cdot \int_r^R (f'(s))^2s^{p}\,ds \le R \int_0^R (f'(s))^2 s^{p}\,ds $$ where the last inequality follows from $r^ps^{-p}\le 1$. Integrate over $r$ to obtain $$\int_0^R r^{p} f(r)^2\,dr\le R^2\int_0^R (f'(s))^2 s^{p}\,ds $$

Compact embedding

It suffices to embed the subspace of functions vanishing at $R$, because this subspace is of codimension $1$ in $W^{1,2}((0,R),r^p)$. Let's change the variables to get rid of the weight in the energy integral. The substitution is $t=r^{1-p}$ if $p>1$, or $t=-\log r$ if $p=1$. Let $g(t)=f(r)$. After the substitution, $$\int_0^1 f'(r)^2 r^p\,dr = \int_a^\infty g'(t)^2\,dt \tag1$$ where $a=R^{1-p}$ if $p>1$, or $t=-\log R$ if $p=1$. Also, $$\int_0^1 f(r)^2 r^p\,dr = \int_a^\infty g(t)^2 w(t)\,dt\tag2$$ where $$w(t)=\begin{cases}Ct^{2p/(1-p)} \quad &p>1 \\ e^{-2t} \quad &p=1 \end{cases}$$

Thus, we can work with the space of functions for which (1) and (2) are finite, and try to embed it into the weighted space $L^2_w$. The important point is that the integral of $w$ is finite. Thanks to this, the compactness proof works almost in the same way as in the unweighted case (e.g., Theorem 11.10 in A first course in Sobolev spaces by Leoni, to which I refer below). The preliminary steps are:

1. extend $g$ and $w$ to $\mathbb R$ by reflection
2. $\int_{\mathbb R}|g(t+h)-g(t)|^2\,dt\le h^2\int_\mathbb R g'(t)^2\,dt$ (Lemma 11.11)
3. $\int_{\mathbb R}|(g*\varphi_\delta)(t)-g(t)|^2\,dt\le C\delta^2\int_\mathbb R g'(t)^2\,dt$ (Lemma 11.12)

Consider a sequence $g_n$ with bounded integrals (1)-(2) and with $g_n(a)=0$. Observe that they are uniformly bounded, thanks to boundedness of (1). Extract a subsequence that weakly converges to some $g$. Using item 3 above, pick $\delta>0$ such that $\int_{\mathbb R}|(g_n*\varphi_\delta)-g_n|^2$ is small (uniformly in $n$). Thus, we can work with mollified functions. But they converge pointwise to $g*\varphi_\delta$, and are uniformly bounded. Since the sequence $|g_n*\varphi_\delta-g*\varphi_\delta|^2 w$ is dominated by a constant multiple of $w$, the dominated convergence theorem yields $$\int_{\mathbb R} |g_n*\varphi_\delta-g*\varphi_\delta|^2 w\to 0\quad \text{as } n\to\infty\qquad \Box$$