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We study the solvability of the regularity problem for degenerate ellipticoperators in the block case for data in weighted spaces. More precisely, let$L_w$ be a degenerate elliptic operator with degeneracy given by a fixed weight$w\in A_2(dx)$ in $\mathbb{R}^n$, and consider the associated block secondorder degenerate elliptic problem in the upper-half space $\mathbb{R}_+^{n+1}$.We obtain non-tangential bounds for the full gradient of the solution of theblock case operator given by the Poisson semigroup in terms of the gradient ofthe boundary data. All this is done in the spaces $L^p(vdw)$ where $v$ is aMuckenhoupt weight with respect to the underlying natural weighted space$(\mathbb{R}^n, wdx)$. We recover earlier results in the non-degenerate case(when $w\equiv 1$, and with or without weight $v$). Our strategy is alsodifferent and more direct thanks in particular to recent observations on changeof angles in weighted square function estimates and non-tangential maximalfunctions. Our method gives as a consequence the (unweighted)$L^2(dx)$-solvability of the regularity problem for the block operator \[\mathbb{L}_\alpha u(x,t) = -|x|^{\alpha} \mathrm{div}_x \big(|x|^{-\alpha}\,A(x) \nabla_x u(x,t)\big)-\partial_{t}^2 u(x,t) \] for any complex-valueduniformly elliptic matrix $A$ and for all $-\epsilon<\alpha<\frac{2\,n}{n+2}$,where $\epsilon$ depends just on the dimension and the ellipticity constants of$A$.

The regularity problem for degenerate elliptic operators in weighted spaces