Second order elliptic operators with complex bounded measurable coefficients in $L^p$, Sobolev and Hardy spaces

Hofmann, Steve; Mayboroda, Svitlana; McIntosh, Alan

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 44 (2011) no. 5, pp. 723-800.

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 $p\notin [2n/(n+2),2n/(n-2\left)\right]$. 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 $[2n/(n+2),2n/(n-2\left)\right]$. 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.

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 $p\notin [2n/(n+2),2n/(n-2\left)\right]$. 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 $[2n/(n+2),2n/(n-2\left)\right]$. 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.

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

[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 ${ℝ}^n$, 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 ${ℝ}^n$ 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 ${ℝ}^n$, 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 ${ℝ}^n$, 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 $H^{\infty }$ 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 $H^{\infty }$-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 $H^p\left({𝐑}^n\right)$ 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 $H_{\infty }$ 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