Second order elliptic operators with complex bounded measurable coefficients in L p , Sobolev and Hardy spaces
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 44 (2011) no. 5, pp. 723-800.

Let L be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with L, such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in L p , Sobolev, and some new Hardy spaces naturally associated to L. First, we show that the known ranges of boundedness in L p for the heat semigroup and Riesz transform of L, are sharp. In particular, the heat semigroup e -tL need not be bounded in L p if p[2n/(n+2),2n/(n-2)]. Then we provide a complete description of all Sobolev spaces in which L admits a bounded functional calculus, in particular, where e -tL is bounded. Secondly, we develop a comprehensive theory of Hardy and Lipschitz spaces associated to L, that serves the range of p beyond [2n/(n+2),2n/(n-2)]. It includes, in particular, characterizations by the sharp maximal function and the Riesz transform (for certain ranges of p), as well as the molecular decomposition and duality and interpolation theorems.

Soit L un opérateur elliptique du second ordre de formes de divergence, à coefficients complexes bornés et mesurables. Les opérateurs associés à L tels que le semi-groupe de la chaleur ou la transformée de Riesz ne sont en général pas de type Calderón-Zygmund et présentent des comportements différents de leurs analogues construits à partir du laplacien. Cet article a pour objectif de décrire de manière exhaustive les propriétés de ces opérateurs dans L p , dans les espaces de Sobolev ainsi que dans certains nouveaux espaces de Hardy naturellement associés à L. Tout d’abord, nous montrons que les plages de valeurs connues pour lesquelles ces opérateurs sont bornés en norme L p sont strictes. En particulier, le semi-groupe de la chaleur et la transformée de Riesz ne sont pas obligatoirement bornés si p[2n/(n+2),2n/(n-2)]. Nous fournissons ensuite une description complète de tous les espaces de Sobolev pour lesquels L admet un calcul fonctionnel borné, en particulier, pour lesquels e -tL est borné. Puis, nous développons une théorie extensive des espaces de Hardy et de Lipschitz associés à L, pour les valeurs de p hors de [2n/(n+2),2n/(n-2)]. Cette théorie comprend, en particulier, des caractérisations par la fonction maximale « dièse » et par la transformée de Riesz (pour certaines plages de p), ainsi que leur décomposition moléculaire, leur dualité et les théorèmes d’interpolation.

DOI: 10.24033/asens.2154
Classification: 42B30, 42B35, 42B25, 35J15
Keywords: Hardy and Lipschitz spaces, elliptic operators, complex coefficients, heat semigroup, Riesz transform
Mot clés : espaces de Hardy et de Lipschitz, opérateurs elliptiques, coefficients complexes, semi-groupe de la chaleur, transformation de Riesz
@article{ASENS_2011_4_44_5_723_0,
     author = {Hofmann, Steve and Mayboroda, Svitlana and McIntosh, Alan},
     title = {Second order elliptic operators with complex bounded measurable coefficients in~$L^p$, {Sobolev} and {Hardy} spaces},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {723--800},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 44},
     number = {5},
     year = {2011},
     doi = {10.24033/asens.2154},
     mrnumber = {2931518},
     zbl = {1243.47072},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2154/}
}
TY  - JOUR
AU  - Hofmann, Steve
AU  - Mayboroda, Svitlana
AU  - McIntosh, Alan
TI  - Second order elliptic operators with complex bounded measurable coefficients in $L^p$, Sobolev and Hardy spaces
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2011
SP  - 723
EP  - 800
VL  - 44
IS  - 5
PB  - Société mathématique de France
UR  - http://www.numdam.org/articles/10.24033/asens.2154/
DO  - 10.24033/asens.2154
LA  - en
ID  - ASENS_2011_4_44_5_723_0
ER  - 
%0 Journal Article
%A Hofmann, Steve
%A Mayboroda, Svitlana
%A McIntosh, Alan
%T Second order elliptic operators with complex bounded measurable coefficients in $L^p$, Sobolev and Hardy spaces
%J Annales scientifiques de l'École Normale Supérieure
%D 2011
%P 723-800
%V 44
%N 5
%I Société mathématique de France
%U http://www.numdam.org/articles/10.24033/asens.2154/
%R 10.24033/asens.2154
%G en
%F ASENS_2011_4_44_5_723_0
Hofmann, Steve; Mayboroda, Svitlana; McIntosh, Alan. Second order elliptic operators with complex bounded measurable coefficients in $L^p$, Sobolev and Hardy spaces. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 44 (2011) no. 5, pp. 723-800. doi : 10.24033/asens.2154. http://www.numdam.org/articles/10.24033/asens.2154/

[1] D. Albrecht, X. Duong & A. Mcintosh, Operator theory and harmonic analysis, in Instructional Workshop on Analysis and Geometry, Part III (Canberra, 1995), Proc. Centre Math. Appl. Austral. Nat. Univ. 34, Austral. Nat. Univ., 1996, 77-136. | MR | Zbl

[2] J. Alvarez & M. Milman, Spaces of Carleson measures: duality and interpolation, Ark. Mat. 25 (1987), 155-174. | MR | Zbl

[3] J. Alvarez & M. Milman, Interpolation of tent spaces and applications, in Function spaces and applications (Lund, 1986), Lecture Notes in Math. 1302, Springer, 1988, 11-21. | MR | Zbl

[4] P. Auscher, Some questions on elliptic operators, in Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), Contemp. Math. 338, Amer. Math. Soc., 2003, 1-10. | MR | Zbl

[5] P. Auscher, On L p estimates for square roots of second order elliptic operators on n , Publ. Mat. 48 (2004), 159-186. | MR | Zbl

[6] P. Auscher, On necessary and sufficient conditions for L p -estimates of Riesz transforms associated to elliptic operators on n and related estimates, Mem. Amer. Math. Soc. 186 (2007). | MR | Zbl

[7] P. Auscher & T. Coulhon, Riesz transform on manifolds and Poincaré inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci. 4 (2005), 531-555. | Numdam | MR | Zbl

[8] P. Auscher, T. Coulhon & P. Tchamitchian, Absence de principe du maximum pour certaines équations paraboliques complexes, Colloq. Math. 71 (1996), 87-95. | MR | Zbl

[9] P. Auscher, X. T. Duong & A. Mcintosh, Boundedness of Banach space valued singular integral operators and Hardy spaces, preprint, 2005.

[10] P. Auscher, S. Hofmann, M. Lacey, A. Mcintosh & P. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on n , Ann. of Math. 156 (2002), 633-654. | MR | Zbl

[11] P. Auscher, A. Mcintosh & E. Russ, Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal. 18 (2008), 192-248. | MR | Zbl

[12] P. Auscher & E. Russ, Hardy spaces and divergence operators on strongly Lipschitz domains of n , J. Funct. Anal. 201 (2003), 148-184. | MR | Zbl

[13] P. Auscher & P. Tchamitchian, Calcul fontionnel précisé pour des opérateurs elliptiques complexes en dimension un (et applications à certaines équations elliptiques complexes en dimension deux), Ann. Inst. Fourier (Grenoble) 45 (1995), 721-778. | EuDML | Numdam | MR | Zbl

[14] P. Auscher & P. Tchamitchian, Square root problem for divergence operators and related topics, Astérisque 249 (1998). | Numdam | MR | Zbl

[15] A. Bernal, Some results on complex interpolation of T q p spaces, in Interpolation spaces and related topics (Haifa, 1990), Israel Math. Conf. Proc. 5, Bar-Ilan Univ., 1992, 1-10. | MR | Zbl

[16] A. Bernal & J. Cerdà, Complex interpolation of quasi-Banach spaces with an A-convex containing space, Ark. Mat. 29 (1991), 183-201. | MR | Zbl

[17] S. Blunck & P. C. Kunstmann, Calderón-Zygmund theory for non-integral operators and the H functional calculus, Rev. Mat. Iberoamericana 19 (2003), 919-942. | EuDML | MR | Zbl

[18] S. Blunck & P. C. Kunstmann, Weak type (p,p) estimates for Riesz transforms, Math. Z. 247 (2004), 137-148. | MR | Zbl

[19] A.-P. Calderón & A. Torchinsky, Parabolic maximal functions associated with a distribution. II, Advances in Math. 24 (1977), 101-171. | MR | Zbl

[20] W. S. Cohn & I. E. Verbitsky, Factorization of tent spaces and Hankel operators, J. Funct. Anal. 175 (2000), 308-329. | MR | Zbl

[21] R. R. Coifman, A real variable characterization of H p , Studia Math. 51 (1974), 269-274. | EuDML | MR | Zbl

[22] R. R. Coifman, Y. Meyer & E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), 304-335. | MR | Zbl

[23] R. R. Coifman & G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645. | MR | Zbl

[24] M. Cwikel, M. Milman & Y. Sagher, Complex interpolation of some quasi-Banach spaces, J. Funct. Anal. 65 (1986), 339-347. | MR | Zbl

[25] E. B. Davies, Limits on L p regularity of self-adjoint elliptic operators, J. Differential Equations 135 (1997), 83-102. | MR | Zbl

[26] X. T. Duong, J. Xiao & L. Yan, Old and new Morrey spaces with heat kernel bounds, J. Fourier Anal. Appl. 13 (2007), 87-111. | MR | Zbl

[27] X. T. Duong & L. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc. 18 (2005), 943-973. | MR | Zbl

[28] X. T. Duong & L. Yan, New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications, Comm. Pure Appl. Math. 58 (2005), 1375-1420. | MR | Zbl

[29] P. L. Duren, B. W. Romberg & A. L. Shields, Linear functionals on H p spaces with 0<p<1, J. reine angew. Math. 238 (1969), 32-60. | EuDML | MR | Zbl

[30] J. Dziubański & M. Preisner, Riesz transform characterization of Hardy spaces associated with Schrödinger operators with compactly supported potentials, Ark. Mat. 48 (2010), 301-310. | MR | Zbl

[31] J. Dziubański & J. Zienkiewicz, Hardy spaces associated with some Schrödinger operators, Studia Math. 126 (1997), 149-160. | EuDML | MR | Zbl

[32] C. Fefferman & E. M. Stein, H p spaces of several variables, Acta Math. 129 (1972), 137-193. | MR | Zbl

[33] M. Frazier & B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), 34-170. | MR | Zbl

[34] J. Frehse, An irregular complex valued solution to a scalar uniformly elliptic equation, Calc. Var. Partial Differential Equations 33 (2008), 263-266. | MR | Zbl

[35] J. García-Cuerva & J. L. Rubio De Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies 116, North-Holland Publishing Co., 1985. | MR | Zbl

[36] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Math. Studies 105, Princeton Univ. Press, 1983. | MR | Zbl

[37] M. E. Gomez & M. Milman, Complex interpolation of H p spaces on product domains, Ann. Mat. Pura Appl. 155 (1989), 103-115. | MR | Zbl

[38] S. Hofmann, G. Lu, D. Mitrea, M. Mitrea & L. Yan, Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, preprint http://www.math.wayne.edu/~gzlu/papers/HLMMY22.pdf. | MR | Zbl

[39] S. Hofmann & J. M. Martell, L p bounds for Riesz transforms and square roots associated to second order elliptic operators, Publ. Mat. 47 (2003), 497-515. | EuDML | MR | Zbl

[40] S. Hofmann & S. Mayboroda, Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann. 344 (2009), 37-116. | MR | Zbl

[41] S. Hofmann & S. Mayboroda, Correction to [40], preprint arXiv:0907.0129. | MR

[42] T. Hytönen, J. Van Neerven & P. Portal, Conical square function estimates in UMD Banach spaces and applications to H -functional calculi, J. Anal. Math. 106 (2008), 317-351. | MR | Zbl

[43] S. Janson & P. W. Jones, Interpolation between H p spaces: the complex method, J. Funct. Anal. 48 (1982), 58-80. | MR | Zbl

[44] R. Jiang & D. Yang, New Orlicz-Hardy spaces associated with divergence form elliptic operators, J. Funct. Anal. 258 (2010), 1167-1224. | MR | Zbl

[45] N. Kalton, S. Mayboroda & M. Mitrea, Interpolation of Hardy-Sobolev-Besov-Triebel-Lizorkin spaces and applications to problems in partial differential equations, in Interpolation theory and applications, Contemp. Math. 445, Amer. Math. Soc., 2007, 121-177. | MR | Zbl

[46] N. Kalton & M. Mitrea, Stability of fredholm properties on interpolation scales of quasi-Banach spaces and applications, Trans. Amer. Math. Soc. 350 (1998), 3837-3901. | MR | Zbl

[47] R. H. Latter, A characterization of H p (𝐑 n ) in terms of atoms, Studia Math. 62 (1978), 93-101. | EuDML | MR | Zbl

[48] J. M. Martell, Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications, Studia Math. 161 (2004), 113-145. | EuDML | MR | Zbl

[49] S. Mayboroda, The connections between Dirichlet, regularity and Neumann problems for second order elliptic operators with complex bounded measurable coefficients, Adv. Math. 225 (2010), 1786-1819. | MR | Zbl

[50] V. G. MazʼYa, S. A. Nazarov & B. A. Plamenevskiĭ, Absence of a De Giorgi-type theorem for strongly elliptic equations with complex coefficients, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 115 (1982), 156-168. | MR | Zbl

[51] A. Mcintosh, Operators which have an H functional calculus, in Miniconference on operator theory and partial differential equations (North Ryde, 1986), Proc. Centre Math. Anal. Austral. Nat. Univ. 14, Austral. Nat. Univ., 1986, 210-231. | MR | Zbl

[52] O. Mendez & M. Mitrea, The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations, J. Fourier Anal. Appl. 6 (2000), 503-531. | EuDML | MR | Zbl

[53] N. G. Meyers, Mean oscillation over cubes and Hölder continuity, Proc. Amer. Math. Soc. 15 (1964), 717-721. | MR | Zbl

[54] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43, Princeton Univ. Press, 1993. | MR | Zbl

[55] E. M. Stein & G. Weiss, On the theory of harmonic functions of several variables 103 (1960), 25-62. | MR | Zbl

[56] M. H. Taibleson & G. Weiss, The molecular characterization of certain Hardy spaces, Astérisque 77 (1980), 67-149. | Numdam | MR | Zbl

[57] H. Triebel, Theory of function spaces, Monographs in Math. 78, Birkhäuser, 1983. | MR | Zbl

[58] T. H. Wolff, A note on interpolation spaces, in Harmonic analysis (Minneapolis, Minn., 1981), Lecture Notes in Math. 908, Springer, 1982, 199-204. | MR | Zbl

[59] L. Yan, Classes of Hardy spaces associated with operators, duality theorem and applications, Trans. Amer. Math. Soc. 360 (2008), 4383-4408. | MR | Zbl

Cited by Sources: