On the Kato problem and extensions for degenerate elliptic operators

Cruz-Uribe, David; Martell, José María; Rios, Cristian

doi:10.2140/apde.2018.11.609

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On the Kato problem and extensions for degenerate elliptic operators

David Cruz-Uribe, José María Martell and Cristian Rios

Vol. 11 (2018), No. 3, 609–660

DOI: 10.2140/apde.2018.11.609

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 $L_{w} = - w^{- 1} div (A \nabla)$ , where $w$ is in the Muckenhoupt class $A_{2}$ and $A$ is a $w$ -degenerate elliptic measure (that is, $A = w B$ with $B (x)$ an $n \times n$ bounded, complex-valued, uniformly elliptic matrix), then $L_{w}$ satisfies the weighted estimate $∥ \sqrt{L_{w}} f ∥_{L^{2} (w)} \approx ∥ \nabla f ∥_{L^{2} (w)}$ . 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:

∥ L_{w}^{1 ∕ 2} f ∥_{L^{2} (ℝ^{n})} \approx ∥ \nabla f ∥_{L^{2} (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 degree of degeneracy in its ellipticity. For example, we consider the family of operators $L_{γ} = - | x |^{γ} div (| x |^{- γ} B (x) \nabla)$ , 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

∥ L_{γ}^{1 ∕ 2} f ∥_{L^{2} (ℝ^{n})} \approx ∥ \nabla f ∥_{L^{2} (ℝ^{n})}, - ϵ < γ < \frac{2 n}{n + 2} .

The case $γ = 0$ corresponds to the case of uniformly elliptic matrices. Hence, our result gives a range of $γ$ ’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 $L^{p} (w)$ , and also on $L^{p} (v d w)$ with $v \in A_{\infty} (w)$ , 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.

Keywords

Muckenhoupt weights, degenerate elliptic operators, Kato problem, semigroups, holomorphic functional calculus, square functions, square roots of elliptic operators, Riesz transforms, Dirichlet problem, regularity problem, Neumann problem

Mathematical Subject Classification 2010

Primary: 35B45, 35J15, 35J25, 35J70, 42B20

Secondary: 42B37, 47A07, 47B44, 47D06

Milestones

Received: 6 October 2016

Revised: 6 June 2017

Accepted: 20 September 2017

Published: 22 November 2017

Authors

David Cruz-Uribe
Department of Mathematics University of Alabama Tuscaloosa, AL United States

José María Martell
Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM Consejo Superior de Investigaciones Científicas Madrid Spain	Department of Mathematics University of Missouri Columbia, MO USA

Cristian Rios
Department of Mathematics and Statistics University of Calgary Canada

Vol. 11, No. 3, 2018