On Harnack inequality and Hölder continuity for non uniformly elliptic equations

We study Harnack inequality and a priori Hölder estimates for weak solutions to a new class of equations 1 \(\begin{aligned} \partial _{z_i} \left( a_{ij}(z)\partial _{z_j} u \right) =0 \end{aligned}\) $$\begin{array}{c}\hfill {\partial }_{z_i}\left(a_{\mathit{ij}},(z),{\partial }_{z_j},u\right)=0\end{array}$$ satisfying the non-uniform ellipticity condition \(\begin{aligned} C_1 \left( \omega (x, t) \vert \xi \vert ^2+\vert \eta \vert ^2 \right) \le a_{ij}(z) \zeta _i \zeta _j \le C_2 \left( \omega (x, t) \vert \xi \vert ^2+\vert \eta \vert ^2 \right) \end{aligned}\) $$\begin{array}{c}\hfill C_1\left({\omega (x,t)|\xi |}^2,+,{|\eta |}^2\right)\le a_{\mathit{ij}}(z){\zeta }_i{\zeta }_j\le C_2\left({\omega (x,t)|\xi |}^2,+,{|\eta |}^2\right)\end{array}$$ where \( \zeta =(\xi , \eta )\in \mathbb {R}^n\times \mathbb {R}^m, \, \zeta \ne 0 \) $\zeta =(\xi ,\eta )\in {\mathbb{R}}^n\times {\mathbb{R}}^m,\zeta \ne 0$ where \( A=\left\{ a_{ij}(z)\right\} _{i,j=1,...N} \) $A={\left\{a_{\mathit{ij}},(z)\right\}}_{i,j=1,...N}$ is a positive matrix defined on a bounded domain \(\Omega \in \mathbb {R}^N \) $\mathrm{\Omega }\in {\mathbb{R}}^N$ of points \(z=(x,t), \, x\in \mathbb {R}^n, \, t\in \mathbb {R}^m, \) $z=(x,t),x\in {\mathbb{R}}^n,t\in {\mathbb{R}}^m,$ \(N=n+m; \, n,m\ge 1.\) $N=n+m;n,m\ge 1.$ The weight \(\omega (x, t)\) $\omega (x,t)$ is a positive function satisfying also some additional conditions. We prove our results by using Sobolev and Poincare-type inequalities modeled on the non-uniformity of the gradient.