# Real interpolation of weighted Sobolev spaces with different weights

Let $\Omega \subseteq \mathbb{R}^n$ be open and let $w_0$ and $w_1$ be measurable and almost everywhere positive and finite functions defined on $\Omega$. Let $L^2_{w_0}(\Omega)$ be the weighted Lebesgue space and $H^1_{w_0}(\Omega)$ be the weighted Sobolev space defined by the norms $\|u\|_{L^2_{w_0}(\Omega)}^2 = \int_\Omega |u(x)|^2 \, w_0(x) \, dx$ and $$\|u\|_{H^1_{w_0}(\Omega)}^2 := \|u\|_{L^2_{w_0}(\Omega)}^2 + \sum_{i=0}^n \left\|\frac{\partial u}{\partial x_i}\right\|_{L^2_{w_0}(\Omega)}^2,$$ respectively. Let $L^2_{w_1}(\Omega)$ and $H^1_{w_1}(\Omega)$ be defined likewise. Under the assumption $w_0^{-1}, w_1^{-1} \in L^1_{\mathrm{loc}}(\Omega)$, $H^1_{w_0}(\Omega)$ and $H^1_{w_1}(\Omega)$ are Hilbert spaces [1]. Given $\theta \in (0,1)$ it is known that $$\left(L^2_{w_0}(\Omega),L^2_{w_1}(\Omega)\right)_{\theta,2} = L^2_{w_0^{1-\theta} w_1^\theta}(\Omega)$$ with equivalence of norms [2, Lem. 23.1], [3, Th. 3.1].

The question:
Given $\theta \in (0,1)$, is there a known characterization of the real interpolation space $$\left(H^1_{w_0}(\Omega),H^1_{w_1}(\Omega)\right)_{\theta,2}$$ in terms of some another weighted Sobolev space, maybe for special classes of domains and weights? I'm particularly interested in the case of $\Omega$ being the unit ball of $\mathbb{R}^n$ and $w_0$ and $w_1$ being powers of the distance to the boundary function.

Related results I found so far:

• In [4] S. G. Pyatkov characterizes $$\left(H^m_{p,\Psi}(\Omega),L_{p,\omega}(\Omega)\right)_{\theta,p},$$ where $\|u\|_{H_{p,\Psi}^m(\Omega)}^p = \int_{\Omega}\sum_{|\alpha|\le m}\omega_{\alpha}\bigl|D^{\alpha}u(x)\bigr|^p\,dx$ and $\|u\|_{L_{p,\omega}(\Omega)}^p=\int_{\Omega}\omega(x)\bigl|u(x)\bigr|^p\,dx$, in terms of some kind of Besov space.

• In [5, §3.3] C. Foiaş and J.-L. Lions prove a result similar to what I want but require that the weights ($M_j$ in their notation) obey a condition of the form $M_j^{-1} \frac{\partial M_j}{\partial x_i} \in L^\infty(\Omega)$, which excludes non-zero powers of the distance to the boundary function.

References:
[1] A. Kufner & B. Opic, How to define reasonably weighted Sobolev spaces, Commentationes Mathematicae Universitatis Carolinae 25(3), 537–554, 1984.

[2] L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lecture Notes of the Unione Matematica Italiana, 3. Springer, Berlin; UMI, Bologna, 2007.

[3] S. N. Chandler-Wilde, D. P. Hewett & A. Moiola. Interpolation of Hilbert and Sobolev spaces: quantitative estimates and counterexamples, Mathematika 61(2), 414–443, 2015. —doi:10.1112/S0025579314000278, arXiv:1404.3599 [math.FA]

[4] S. G. Pyatkov. Interpolation of Weighted Sobolev Spaces (in Russian), Matematicheskie Trudy 4(1), 122–173, 2001. —mi.mathnet.ru/mt8

[5] C. Foiaş & J.-L. Lions, Sur certains théorèmes d'interpolation (in French) Acta Universitatis Szegediensis. Acta Scientiarum Mathematicarum 22(3-4), 269–282, 1961.