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Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. (English) Zbl 1078.42013
The main purpose of this paper is to prove a generalization of Fefferman and Stein’s result on the duality of $$H^1$$ and BMO spaces. In a previous paper of one of the authors, a class of new Hardy spaces $$H^1_L$$ were introduced where $$L$$ is the infinitesimal generator of an analytic semigroup $$\{e^{-tL}\}_{t>0}$$ on $$L^2$$ with kernel $$p_t(x,y)$$. Here they prove that if $$L$$ has a bounded holomorphic functional calculus on $$L^2$$ and the kernel $$p_t$$ satisfies an upper bound of Poisson type, then the spaces $$\text{BMO}_{L^*}$$ is the dual space of the Hardy space $$H^1_L$$ where $$L^*$$ denotes the adjoint operator of $$L$$ and $$\text{BMO}_{L}$$ is defined as the set of functions $$f$$ such that $\sup_{B}{1\over | B| } \int_B | f(x)-P_{t_B}f(x)| dx <\infty$ and $P_{t_B}f(x)=\int_{\mathbb R^n} p_{t_B}(x,y) f(y) dy$ with $$t_B$$ the radius of the ball $$B$$. They also obtain a characterization of $$\text{BMO}_L$$ in terms of Carleson measures and give a sufficient condition for the classical BMO space and $$\text{BMO}_L$$ to coincide.

##### MSC:
 42B30 $$H^p$$-spaces 42B35 Function spaces arising in harmonic analysis
##### Keywords:
Hardy spaces; BMO; semigroup of operators; duality; Carleson measures
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##### References:
 [1] Pascal Auscher, Steve Hofmann, John L. Lewis, and Philippe Tchamitchian, Extrapolation of Carleson measures and the analyticity of Kato’s square-root operators, Acta Math. 187 (2001), no. 2, 161 – 190. , https://doi.org/10.1007/BF02392615 Steve Hofmann, Michael Lacey, and Alan McIntosh, The solution of the Kato problem for divergence form elliptic operators with Gaussian heat kernel bounds, Ann. of Math. (2) 156 (2002), no. 2, 623 – 631. , https://doi.org/10.2307/3597200 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 \Bbb R$$^{n}$$, Ann. of Math. (2) 156 (2002), no. 2, 633 – 654. · Zbl 1128.35316 · doi:10.2307/3597201 · doi.org [2] P. Auscher and 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), no. 3, 721 – 778 (French, with English and French summaries). · Zbl 0819.35028 [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). · Zbl 0909.35001 [4] P. Auscher, X.T. Duong and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces, preprint, 2004. [5] Thierry Coulhon and Xuan Thinh Duong, Maximal regularity and kernel bounds: observations on a theorem by Hieber and Prüss, Adv. Differential Equations 5 (2000), no. 1-3, 343 – 368. · Zbl 1001.34046 [6] Michael Cowling, Ian Doust, Alan McIntosh, and Atsushi Yagi, Banach space operators with a bounded \?^\infty functional calculus, J. Austral. Math. Soc. Ser. A 60 (1996), no. 1, 51 – 89. · Zbl 0853.47010 [7] R. R. Coifman, Y. Meyer, and E. M. Stein, Un nouvel éspace fonctionnel adapté à l’étude des opérateurs définis par des intégrales singulières, Harmonic analysis (Cortona, 1982) Lecture Notes in Math., vol. 992, Springer, Berlin, 1983, pp. 1 – 15 (French). · doi:10.1007/BFb0069149 · doi.org [8] R. R. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), no. 2, 304 – 335. · Zbl 0569.42016 · doi:10.1016/0022-1236(85)90007-2 · doi.org [9] Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569 – 645. · Zbl 0358.30023 [10] E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989. · Zbl 0699.35006 [11] D. G. Deng, On a generalized Carleson inequality, Studia Math. 78 (1984), no. 3, 245 – 251. · Zbl 0558.42011 [12] D.G. Deng, X.T. Duong, A. Sikora and L.X. Yan, Comparison between the classical BMO and the BMO spaces associated with operators and applications, preprint, (2005). · Zbl 1283.42036 [13] J. Dziubański, G. Garrigós, T. Martínez, J. L. Torrea, and J. Zienkiewicz, \?\?\? spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality, Math. Z. 249 (2005), no. 2, 329 – 356. · Zbl 1136.35018 · doi:10.1007/s00209-004-0701-9 · doi.org [14] Xuan Thinh Duong and Alan MacIntosh, Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iberoamericana 15 (1999), no. 2, 233 – 265. , https://doi.org/10.4171/RMI/255 Xuan Thinh Duong and Alan McIntosh, Corrigenda: ”Singular integral operators with non-smooth kernels on irregular domains”, Rev. Mat. Iberoamericana 16 (2000), no. 1, 217. · Zbl 1056.42506 · doi:10.4171/RMI/273 · doi.org [15] Xuan T. Duong and Derek W. Robinson, Semigroup kernels, Poisson bounds, and holomorphic functional calculus, J. Funct. Anal. 142 (1996), no. 1, 89 – 128. · Zbl 0932.47013 · doi:10.1006/jfan.1996.0145 · doi.org [16] X.T. Duong and L.X. Yan, New function spaces of BMO type, the John-Nirenberg inequality, interpolation and applications, to appear, Comm. Pure Appl. Math., (2005). · Zbl 1153.26305 [17] Jacek Dziubański and Jacek Zienkiewicz, Hardy spaces associated with some Schrödinger operators, Studia Math. 126 (1997), no. 2, 149 – 160. · Zbl 0918.42013 [18] Charles Fefferman, Characterizations of bounded mean oscillation, Bull. Amer. Math. Soc. 77 (1971), 587 – 588. · Zbl 0229.46051 [19] C. Fefferman and E. M. Stein, \?^\? spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137 – 193. · Zbl 0257.46078 · doi:10.1007/BF02392215 · doi.org [20] Steve Hofmann and José María Martell, \?^\? bounds for Riesz transforms and square roots associated to second order elliptic operators, Publ. Mat. 47 (2003), no. 2, 497 – 515. · Zbl 1074.35031 · doi:10.5565/PUBLMAT_47203_12 · doi.org [21] Jean-Lin Journé, Calderón-Zygmund operators, pseudodifferential operators and the Cauchy integral of Calderón, Lecture Notes in Mathematics, vol. 994, Springer-Verlag, Berlin, 1983. [22] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415 – 426. · Zbl 0102.04302 · doi:10.1002/cpa.3160140317 · doi.org [23] Peter Li, Harmonic sections of polynomial growth, Math. Res. Lett. 4 (1997), no. 1, 35 – 44. · Zbl 0880.53039 · doi:10.4310/MRL.1997.v4.n1.a4 · doi.org [24] Peter Li and Jiaping Wang, Counting dimensions of \?-harmonic functions, Ann. of Math. (2) 152 (2000), no. 2, 645 – 658. · Zbl 0996.35020 · doi:10.2307/2661394 · doi.org [25] José María Martell, Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications, Studia Math. 161 (2004), no. 2, 113 – 145. · Zbl 1044.42019 · doi:10.4064/sm161-2-2 · doi.org [26] Alan McIntosh, Operators which have an \?_\infty functional calculus, Miniconference on operator theory and partial differential equations (North Ryde, 1986) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 14, Austral. Nat. Univ., Canberra, 1986, pp. 210 – 231. · Zbl 0634.47016 [27] Stephen Semmes, Square function estimates and the \?(\?) theorem, Proc. Amer. Math. Soc. 110 (1990), no. 3, 721 – 726. · Zbl 0719.42023 [28] Zhong Wei Shen, \?^\? estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 2, 513 – 546 (English, with English and French summaries). · Zbl 0818.35021 [29] Zhongwei Shen, On fundamental solutions of generalized Schrödinger operators, J. Funct. Anal. 167 (1999), no. 2, 521 – 564. · Zbl 0936.35051 · doi:10.1006/jfan.1999.3455 · doi.org [30] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501 [31] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. · Zbl 0821.42001 [32] Elias M. Stein and Guido Weiss, On the theory of harmonic functions of several variables. I. The theory of \?^\?-spaces, Acta Math. 103 (1960), 25 – 62. · Zbl 0097.28501 · doi:10.1007/BF02546524 · doi.org [33] Walter A. Strauss, Partial differential equations, John Wiley & Sons, Inc., New York, 1992. An introduction. · Zbl 0817.35001 [34] Alberto Torchinsky, Real-variable methods in harmonic analysis, Pure and Applied Mathematics, vol. 123, Academic Press, Inc., Orlando, FL, 1986. · Zbl 0621.42001 [35] Lixin Yan, Littlewood-Paley functions associated to second order elliptic operators, Math. Z. 246 (2004), no. 4, 655 – 666. · Zbl 1067.42013 · doi:10.1007/s00209-003-0606-z · doi.org [36] Kôsaku Yosida, Functional analysis, 5th ed., Springer-Verlag, Berlin-New York, 1978. Grundlehren der Mathematischen Wissenschaften, Band 123. [37] Yueping Zhu, Area functions on Hardy spaces associated to Schrödinger operators, Acta Math. Sci. Ser. B Engl. Ed. 23 (2003), no. 4, 521 – 530. · Zbl 1047.42017
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