On the Kato problem and extensions for degenerate elliptic operators

Abstract

We study the Kato problem for divergence form operators whose ellipticity may be degenerate. The study of the Kato conjecture for degenerate elliptic equations was begun by Cruz-Uribe and Rios (2008, 2012, 2015). In these papers the authors proved that given an operator $L_w=-w^{-1}div\left(A\nabla \right)$, where $w$ is in the Muckenhoupt class $A_2$ and $A$ is a $w$-degenerate elliptic measure (that is, $A=wB$ with $B\left(x\right)$ an $n\times n$ bounded, complex-valued, uniformly elliptic matrix), then $L_w$ satisfies the weighted estimate $\parallel \sqrt{L_w}f{\parallel }_{L^2\left(w\right)}\approx \parallel \nabla f{\parallel }_{L^2\left(w\right)}$. In the present paper we solve the $L^2$-Kato problem for a family of degenerate elliptic operators. We prove that under some additional conditions on the weight $w$, the following unweighted $L^2$-Kato estimates hold:

$$\parallel L_w^{1∕2}f{\parallel }_{L^2\left({ℝ}^n\right)}\approx \parallel \nabla f{\parallel }_{L^2\left({ℝ}^n\right)}.$$

This extends the celebrated solution to the Kato conjecture by Auscher, Hofmann, Lacey, McIntosh, and Tchamitchian, allowing the differential operator to have some degree of degeneracy in its ellipticity. For example, we consider the family of operators $L_{\gamma }=-|x{|}^{\gamma }div\left(|x{|}^{-\gamma }B\left(x\right)\nabla \right)$, where $B$ is any bounded, complex-valued, uniformly elliptic matrix. We prove that there exists $ϵ>0$, depending only on dimension and the ellipticity constants, such that

$$\parallel L_{\gamma }^{1∕2}f{\parallel }_{L^2\left({ℝ}^n\right)}\approx \parallel \nabla f{\parallel }_{L^2\left({ℝ}^n\right)},-ϵ<\gamma <\frac{2n}{n+2}.$$

The case $\gamma =0$ corresponds to the case of uniformly elliptic matrices. Hence, our result gives a range of $\gamma$’s for which the classical Kato square root proved in Auscher et al. (2002) is an interior point.

Our main results are obtained as a consequence of a rich Calderón–Zygmund theory developed for certain operators naturally associated with $L_w$. These results, which are of independent interest, establish estimates on $L^p\left(w\right)$, and also on $L^p\left(vdw\right)$ with $v\in A_{\infty }\left(w\right)$, for the associated semigroup, its gradient, the functional calculus, the Riesz transform, and vertical square functions. As an application, we solve some unweighted $L^2$-Dirichlet, regularity and Neumann boundary value problems for degenerate elliptic operators.