On the Kato problem and extensions for degenerate elliptic operators

We study the Kato problem for degenerate divergence form operators. This was begun by Cruz-Uribe and Rios who proved that given an operator $L_w=-w^{-1}{\rm div}(A\nabla)$, where $w\in A_2$ and $A$ is a $w$-degenerate elliptic measure (i.e, $A=w\,B$ with $B$ an $n\times n$ bounded, complex-valued, uniformly elliptic matrix), then $L_w$ satisfies the weighted estimate $\|\sqrt{L_w}f\|_{L^2(w)}\approx\|\nabla f\|_{L^2(w)}$. Here we solve the $L^2$-Kato problem: under some additional conditions on the weight $w$, the following unweighted $L^2$-Kato estimates hold $$\|L_w^{1/2}f\|_{L^2(\mathbb{R}^n)}\approx\|\nabla f\|_{L^2(\mathbb{R}^n)}.$$ This extends the celebrated solution to the Kato conjecture by Auscher, Hofmann, Lacey, McIntosh, and Tchamitchian, allowing the differential operator to have some degeneracy in its ellipticity. For example, we consider the family of operators $L_\gamma=-|x|^{\gamma}{\rm div}(|x|^{-\gamma}B(x)\nabla)$, where $B$ is any bounded, complex-valued, uniformly elliptic matrix. We prove that there exists $\epsilon>0$, depending only on dimension and the ellipticity constants, such that $$\|L_\gamma^{1/2}f\|_{L^2(\mathbb{R}^n)}\approx\|\nabla f\|_{L^2(\mathbb{R}^n)}, \qquad -\epsilon<\gamma<\frac{2\,n}{n+2}.$$ This gives a range of $\gamma$'s for which the classical Kato square root $\gamma=0$ is an interior point. Our main results are obtained as a consequence of a rich Calder\'on-Zygmund theory developed for some operators associated with $L_w$. These results, which are of independent interest, establish estimates on $L^p(w)$, and also on $L^p(v\,dw)$ with $v\in A_\infty(w)$, for the associated semigroup, its gradient, the functional calculus, the Riesz transform, and square functions. As an application, we solve some unweighted $L^2$-Dirichlet, Regularity and Neumann boundary value problems for degenerate elliptic operators.