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The Kato problem for operators with weighted ellipticity

Authors: David Cruz-Uribe SFO and Cristian Rios
Journal: Trans. Amer. Math. Soc. 367 (2015), 4727-4756
MSC (2010): Primary 35J15, 35J25, 35J70, 35D30, 47D06, 35B30, 31B10, 35B45
Published electronically: March 2, 2015
MathSciNet review: 3335399
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Abstract: We consider second order operators $ \mathcal {L}_{w}=-w^{-1}\operatorname {div}\mathbf {A}_{w}\nabla $ with ellipticity controlled by a Muckemphout $ A_{2}$ weight $ w $. We prove that the Kato square root estimate $ \left \Vert \mathcal {L} _{w}^{1/2}f\right \Vert _{L^{2}\left ( w\right ) }\approx \left \Vert \nabla f\right \Vert _{L^{2}\left ( w\right ) }$ holds in the weighted space $ L^{2}\left ( w\right ) $.

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  • [1] Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Ph. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on $ {\mathbb{R}}^n$, Ann. of Math. (2) 156 (2002), no. 2, 633-654. MR 1933726 (2004c:47096c),
  • [2] Pascal Auscher and José María Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. II. Off-diagonal estimates on spaces of homogeneous type, J. Evol. Equ. 7 (2007), no. 2, 265-316. MR 2316480 (2008m:47059),
  • [3] Pascal Auscher and Philippe Tchamitchian, Square root problem for divergence operators and related topics, Astérisque 249 (1998), viii+172 (English, with English and French summaries). MR 1651262 (2000c:47092)
  • [4] Filippo Chiarenza and Michelangelo Franciosi, Quasiconformal mappings and degenerate elliptic and parabolic equations, Matematiche (Catania) 42 (1987), no. 1-2, 163-170 (1989). MR 1030914 (91b:35019)
  • [5] Filippo Chiarenza and Raul Serapioni, A remark on a Harnack inequality for degenerate parabolic equations, Rend. Sem. Mat. Univ. Padova 73 (1985), 179-190. MR 799906 (87a:35110)
  • [6] David Cruz-Uribe and Cristian Rios, Gaussian bounds for degenerate parabolic equations, J. Funct. Anal. 255 (2008), no. 2, 283-312. MR 2419963 (2009k:35162),
  • [7] David Cruz-Uribe and Cristian Rios, The solution of the Kato problem for degenerate elliptic operators with Gaussian bounds, Trans. Amer. Math. Soc. 364 (2012), no. 7, 3449-3478. MR 2901220,
  • [8] Nelson Dunford and B. J. Pettis, Linear operations on summable functions, Trans. Amer. Math. Soc. 47 (1940), 323-392. MR 0002020 (1,338b)
  • [9] Eugene B. Fabes, Carlos E. Kenig, and Raul P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), no. 1, 77-116. MR 643158 (84i:35070),
  • [10] Michael Frazier, Björn Jawerth, and Guido Weiss, Littlewood-Paley theory and the study of function spaces, CBMS Regional Conference Series in Mathematics, vol. 79, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1991. MR 1107300 (92m:42021)
  • [11] F. W. Gehring, The $ L^{p}$-integrability of the partial derivatives of quasiconformal mapping, Bull. Amer. Math. Soc. 79 (1973), 465-466. MR 0320308 (47 #8847)
  • [12] Markus Haase, The functional calculus for sectorial operators, Operator Theory: Advances and Applications, vol. 169, Birkhäuser Verlag, Basel, 2006. MR 2244037 (2007j:47030)
  • [13] Tosio Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan 13 (1961), 246-274. MR 0138005 (25 #1453)
  • [14] C. Kenig, Featured review: The solution of the Kato square root problem for second order elliptic operators on $ {\mathbb{R}}\sp n$, Mathematical Reviews MR1933726 (2004c:47096c) (2004).
  • [15] J.-L. Lions, Espaces d'interpolation et domaines de puissances fractionnaires d'opérateurs, J. Math. Soc. Japan 14 (1962), 233-241 (French). MR 0152878 (27 #2850)
  • [16] Alan McIntosh, Square roots of operators and applications to hyperbolic PDE, (Canberra, 1983) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 5, Austral. Nat. Univ., Canberra, 1984, pp. 124-136. MR 757577 (85h:47016)
  • [17] Alan McIntosh and Atsushi Yagi, Operators of type $ \omega $ without a bounded $ H_\infty $ functional calculus, Miniconference on Operators in Analysis (Sydney, 1989) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 24, Austral. Nat. Univ., Canberra, 1990, pp. 159-172. MR 1060121 (91e:47013)
  • [18] Nicholas Miller, Weighted Sobolev spaces and pseudodifferential operators with smooth symbols, Trans. Amer. Math. Soc. 269 (1982), no. 1, 91-109. MR 637030 (83f:47036),
  • [19] El Maati Ouhabaz, Analysis of heat equations on domains, London Mathematical Society Monographs Series, vol. 31, Princeton University Press, Princeton, NJ, 2005. MR 2124040 (2005m:35001)

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Additional Information

David Cruz-Uribe SFO
Affiliation: Department of Mathematics, Trinity College, Hartford, Connecticut 06106

Cristian Rios
Affiliation: Department of Mathematics, University of Calgary, Calgary, Alberta, Canada T2N 1N4

Keywords: Elliptic operators, Kato problem, weighted norm inequalities, functional calculus
Received by editor(s): September 24, 2012
Received by editor(s) in revised form: January 6, 2013, March 9, 2013, March 10, 2013, March 11, 2013, and March 13, 2013
Published electronically: March 2, 2015
Additional Notes: The first author was partially supported by the Stewart-Dorwart faculty development fund at Trinity College
The second author was supported by the Natural Sciences and Engineering Research Council of Canada
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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