Analysis & PDE

  • Anal. PDE
  • Volume 11, Number 3 (2018), 609-660.

On the Kato problem and extensions for degenerate elliptic operators

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

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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:

L w 1 2 f L 2 ( n ) f L 2 ( 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

L γ 1 2 f L 2 ( n ) f L 2 ( n ) , ϵ < γ < 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 Lw. These results, which are of independent interest, establish estimates on Lp(w), and also on Lp(vdw) with vA(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.

Article information

Anal. PDE, Volume 11, Number 3 (2018), 609-660.

Received: 6 October 2016
Revised: 6 June 2017
Accepted: 20 September 2017
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B45: A priori estimates 35J15: Second-order elliptic equations 35J25: Boundary value problems for second-order elliptic equations 35J70: Degenerate elliptic equations 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 42B37: Harmonic analysis and PDE [See also 35-XX] 47A07: Forms (bilinear, sesquilinear, multilinear) 47B44: Accretive operators, dissipative operators, etc. 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]

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


Cruz-Uribe, David; Martell, José María; Rios, Cristian. On the Kato problem and extensions for degenerate elliptic operators. Anal. PDE 11 (2018), no. 3, 609--660. doi:10.2140/apde.2018.11.609.

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