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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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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  • T. Alberico, A. Cianchi, and C. Sbordone, “Fractional integrals and $A_p$-weights: a sharp estimate”, C. R. Math. Acad. Sci. Paris 347:21-22 (2009), 1265–1270.
  • P. Auscher, On necessary and sufficient conditions for $L^p$-estimates of Riesz transforms associated to elliptic operators on $\mathbb{R}^n$ and related estimates, Mem. Amer. Math. Soc. 871, American Mathematical Society, Providence, RI, 2007.
  • P. Auscher and T. Coulhon, “Riesz transform on manifolds and Poincaré inequalities”, Ann. Sc. Norm. Super. Pisa Cl. Sci. $(5)$ 4:3 (2005), 531–555.
  • P. Auscher and J. M. Martell, “Weighted norm inequalities, off-diagonal estimates and elliptic operators, III: Harmonic analysis of elliptic operators”, J. Funct. Anal. 241:2 (2006), 703–746.
  • P. Auscher and J. M. Martell, “Weighted norm inequalities, off-diagonal estimates and elliptic operators, I: General operator theory and weights”, Adv. Math. 212:1 (2007), 225–276.
  • P. Auscher and J. M. Martell, “Weighted norm inequalities, off-diagonal estimates and elliptic operators, II: Off-diagonal estimates on spaces of homogeneous type”, J. Evol. Equ. 7:2 (2007), 265–316.
  • P. Auscher and J. M. Martell, “Weighted norm inequalities, off-diagonal estimates and elliptic operators, IV: Riesz transforms on manifolds and weights”, Math. Z. 260:3 (2008), 527–539.
  • P. Auscher and P. Tchamitchian, Square root problem for divergence operators and related topics, Astérisque 249, Société Mathématique de France, Paris, 1998.
  • P. Auscher, S. Hofmann, M. Lacey, A. McIntosh, and P. Tchamitchian, “The solution of the Kato square root problem for second order elliptic operators on ${\mathbb R}^n$”, Ann. of Math. $(2)$ 156:2 (2002), 633–654.
  • P. Auscher, A. Rosén, and D. Rule, “Boundary value problems for degenerate elliptic equations and systems”, Ann. Sci. Éc. Norm. Supér. $(4)$ 48:4 (2015), 951–1000.
  • F. Bernicot and J. Zhao, “New abstract Hardy spaces”, J. Funct. Anal. 255:7 (2008), 1761–1796.
  • A. Björn and J. Björn, Nonlinear potential theory on metric spaces, EMS Tracts in Mathematics 17, European Mathematical Society, Zürich, 2011.
  • L. Chen, J. M. Martell, and C. Prisuelos-Arribas, “Conical square functions for degenerate elliptic operators”, preprint, 2016. To appear in Adv. Cal. Var.
  • M. Cowling, I. Doust, A. McIntosh, and A. Yagi, “Banach space operators with a bounded $H^\infty$ functional calculus”, J. Austral. Math. Soc. Ser. A 60:1 (1996), 51–89.
  • D. Cruz-Uribe and C. J. Neugebauer, “The structure of the reverse Hölder classes”, Trans. Amer. Math. Soc. 347:8 (1995), 2941–2960.
  • D. Cruz-Uribe and C. Rios, “Gaussian bounds for degenerate parabolic equations”, J. Funct. Anal. 255:2 (2008), 283–312. Correction in 267:9 (2014), 3507–3513.
  • D. Cruz-Uribe and C. Rios, “The solution of the Kato problem for degenerate elliptic operators with Gaussian bounds”, Trans. Amer. Math. Soc. 364:7 (2012), 3449–3478.
  • D. Cruz-Uribe and C. Rios, “The Kato problem for operators with weighted ellipticity”, Trans. Amer. Math. Soc. 367:7 (2015), 4727–4756.
  • J. Duoandikoetxea, Fourier analysis, Graduate Studies in Mathematics 29, American Mathematical Society, Providence, RI, 2001.
  • E. B. Fabes, C. E. Kenig, and R. P. Serapioni, “The local regularity of solutions of degenerate elliptic equations”, Comm. Partial Differential Equations 7:1 (1982), 77–116.
  • J. García-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies 116, North-Holland, Amsterdam, 1985.
  • M. Haase, The functional calculus for sectorial operators, Operator Theory: Advances and Applications 169, Birkhäuser, Basel, 2006.
  • S. Hofmann, P. Le, and A. Morris, “Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations”, preprint, 2015.
  • T. Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften 132, Springer, 1966.
  • P. Le, “$L^p$ bounds of Riesz transform and vertical square functions for degenerate elliptic operators”, preprint, 2015.
  • A. McIntosh, “Operators which have an $H_\infty$ functional calculus”, pp. 210–231 in Miniconference on operator theory and partial differential equations (North Ryde, 1986), edited by B. Jefferies et al., Proc. Centre Math. Anal. Austral. Nat. Univ. 14, Austral. Nat. Univ., Canberra, 1986.
  • N. Miller, “Weighted Sobolev spaces and pseudodifferential operators with smooth symbols”, Trans. Amer. Math. Soc. 269:1 (1982), 91–109.
  • C. Pérez, “Calderón–Zygmund theory related to Poincaré–Sobolev inequalities, fractional integrals and singular integral operators”, in Function spaces: lectures notes of the spring school on analysis (Paseky nad Jizerou, Czech Republic, 1999), edited by J. Lukes and L. Pick, Matfyzpress, Prague, 1999.
  • D. Yang and J. Zhang, “Weighted $L^p$ estimates of Kato square roots associated to degenerate elliptic operators”, Publ. Mat. 61:2 (2017), 395–444.