Vol. 11, No. 3, 2018

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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
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 Lw = w1 div(A), where w is in the Muckenhoupt class A2 and A is a w-degenerate elliptic measure (that is, A = wB with B(x) an n × n bounded, complex-valued, uniformly elliptic matrix), then Lw satisfies the weighted estimate LwfL2(w) fL2(w). In the present paper we solve the L2-Kato problem for a family of degenerate elliptic operators. We prove that under some additional conditions on the weight w, the following unweighted L2-Kato estimates hold:

Lw12f L2(n) fL2(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 Lγ = |x|γ div(|x|γB(x)), 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

Lγ12f L2(n) fL2(n), ϵ < γ < 2n 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 Lw. These results, which are of independent interest, establish estimates on Lp(w), and also on Lp(vdw) with v A(w), for the associated semigroup, its gradient, the functional calculus, the Riesz transform, and vertical square functions. As an application, we solve some unweighted L2-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