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

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

 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]

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

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