Abstract
Consider a second order divergence form elliptic operator L with complex bounded measurable coefficients. In general, operators based on L, such as the Riesz transform or square function, may lie beyond the scope of the Calderón–Zygmund theory. They need not be bounded in the classical Hardy, BMO and even some L p spaces. In this work we develop a theory of Hardy and BMO spaces associated to L, which includes, in particular, a molecular decomposition, maximal and square function characterizations, duality of Hardy and BMO spaces, and a John–Nirenberg inequality.
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Albrecht, D., Duong, X., McIntosh, A.: Operator theory and harmonic analysis. Instructional Workshop on Analysis and Geometry, Part III (Canberra, 1995), pp. 77–136, Proc. Centre Math. Appl. Austral. Nat. Univ., 34, Austral. Nat. Univ., Canberra (1996)
Auscher, P.: Personal communication
Auscher, P.: On necessary and sufficient conditions for L p-estimates of Riesz transforms associated to elliptic operators on \({\mathbb{R}^n}\) and related estimates. Memoirs of the Amer. Math. Soc. 186(871) (2007)
Auscher P., Coulhon T., Tchamitchian Ph.: Absence de principe du maximum pour certaines équations paraboliques complexes. Coll. Math. 171, 87–95 (1996)
Auscher P., Hofmann S., Lacey M., McIntosh A., Tchamitchian Ph.: The solution of the Kato square root problem for second order elliptic operators on \({\mathbb{R}^n}\) . Ann. Math. (2) 156(2), 633–654 (2002)
Auscher P., Hofmann S., Lewis J.L., Tchamitchian Ph.: Extrapolation of Carleson measures and the analyticity of Kato’s square-root operators. Acta Math. 187(2), 161–190 (2001)
Auscher, P., McIntosh, A., Russ, E.: preprint
Auscher P., Russ E.: Hardy spaces and divergence operators on strongly Lipschitz domains of \({\mathbb{R}^n}\) . J. Funct. Anal. 201(1), 148–184 (2003)
Blunck S., Kunstmann P.: Weak-type (p, p) estimates for Riesz transforms. Math. Z. 247(1), 137–148 (2004)
Blunck S., Kunstmann P.: Caldern–Zygmund theory for non-integral operators and the H ∞ functional calculus. Rev. Mat. Iberoamericana 19(3), 919–942 (2003)
Coifman R.R.: A real variable characterization of H p. Studia Math. 51, 269–274 (1974)
Coifman R.R., Meyer Y., Stein E.M.: Some new function spaces and their applications to harmonic analysis. J. Funct. Anal. 62(2), 304–335 (1985)
Coifman R.R., Weiss G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83(4), 569–645 (1977)
Davies E.B.: Limits on L p regularity of selfadjoint elliptic operators. J. Differ. Equ. 135(1), 83–102 (1997)
Duong X.T., Yan L.X.: Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J. Am. Math. Soc. 18, 943–973 (2005)
Duong X.T., Yan L.X.: New function spaces of BMO type, the John–Nirenberg inequality, interpolation, and applications. Comm. Pure Appl. Math. 58(10), 1375–1420 (2005)
Fefferman C., Stein E.M.: H p spaces of several variables. Acta Math. 129(3–4), 137–193 (1972)
Garcia-Cuerva J., Rubio de Francia J.L.: Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies, vol. 116. North-Holland Publishing Co., Amsterdam (1985)
Hofmann S., Martell J.M.: L p bounds for Riesz transforms and square roots associated to second order elliptic operators. Publ. Math. 47(2), 497–515 (2003)
John F., Nirenberg L.: On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14, 415–426 (1961)
Kenig C.E., Pipher J.: The Neumann problem for elliptic equations with nonsmooth coefficients. Invent. Math. 113(3), 447–509 (1993)
Latter R.H.: A characterization of H p(\({\mathbb{R}^n}\)) in terms of atoms. Studia Math. 62(1), 93–101 (1978)
Maz’ya, V.G., Nazarov, S.A., Plamenevskiiĭ, B.A.: Absence of a De Giorgi-type theorem for strongly elliptic equations with complex coefficients. Boundary value problems of mathematical physics and related questions in the theory of functions, 14. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 115 (1982), 156–168, 309
Stein E.M.: Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, NJ (1970)
Stein E.M.: Harmonic analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ (1993)
Stein E.M.: Maximal functions. II. Homogeneous curves. Proc. Natl. Acad. Sci. U.S.A. 73(7), 2176–2177 (1976)
Stein E.M., Weiss G.: On the theory of harmonic functions of several variables. I. The theory of H p-spaces. Acta Math. 103, 25–62 (1960)
Taibleson, M., Weiss, G.: The molecular characterization of certain Hardy spaces. Representation theorems for Hardy spaces. pp. 67–149, Astérisque, 77, Soc. Math. France, Paris (1980)
Wilson J.M.: On the atomic decomposition for Hardy spaces. Pac. J. Math. 116(1), 201–207 (1985)
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S. Hofmann was supported by the National Science Foundation.
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Hofmann, S., Mayboroda, S. Hardy and BMO spaces associated to divergence form elliptic operators. Math. Ann. 344, 37–116 (2009). https://doi.org/10.1007/s00208-008-0295-3
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DOI: https://doi.org/10.1007/s00208-008-0295-3