Hardy spaces meet harmonic weights. (English) Zbl 1496.42030
Summary: We investigate the Hardy space \(H^1_L\) associated with a self-adjoint operator \(L\) defined in a general setting by S. Hofmann et al. [Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates. Providence, RI: American Mathematical Society (AMS) (2011; Zbl 1232.42018)]. We assume that there exists an \(L\)-harmonic non-negative function \(h\) such that the semigroup \(\exp (-tL)\), after applying the Doob transform related to \(h\), satisfies the upper and lower Gaussian estimates. Under this assumption we describe an illuminating characterisation of the Hardy space \(H^1_L\) in terms of a simple atomic decomposition associated with the \(L\)-harmonic function \(h\). Our approach also yields a natural characterisation of the \(BMO\)-type space corresponding to the operator \(L\) and dual to \(H^1_L\) in the same circumstances.
The applications include surprisingly wide range of operators, such as: Laplace operators with Dirichlet boundary conditions on some domains in \({\mathbb{R}^n} \), Schrödinger operators with certain potentials, and Bessel operators.
The applications include surprisingly wide range of operators, such as: Laplace operators with Dirichlet boundary conditions on some domains in \({\mathbb{R}^n} \), Schrödinger operators with certain potentials, and Bessel operators.
MSC:
| 42B30 | \(H^p\)-spaces |
| 42B35 | Function spaces arising in harmonic analysis |
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 47B38 | Linear operators on function spaces (general) |
Keywords:
Hardy space; harmonic weight; atomic decomposition; maximal function; Lusin function; Littlewood-Paley function; non-negative self-adjoint operator; Gaussian bounds; Doob transformCitations:
Zbl 1232.42018References:
| [1] | P. Auscher, X. T. Duong, and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces, Unpublished preprint, 2005. |
| [2] | Auscher, Pascal, Calder\'{o}n reproducing formulas and applications to Hardy spaces, Rev. Mat. Iberoam., 865-900 (2015) · Zbl 1333.42041 · doi:10.4171/RMI/857 |
| [3] | Auscher, Pascal, Hardy spaces of differential forms and Riesz transforms on Riemannian manifolds, C. R. Math. Acad. Sci. Paris, 103-108 (2007) · Zbl 1109.43005 · doi:10.1016/j.crma.2006.11.023 |
| [4] | Auscher, Pascal, Hardy spaces and divergence operators on strongly Lipschitz domains of \(\mathbb{R}^n\), J. Funct. Anal., 148-184 (2003) · Zbl 1033.42019 · doi:10.1016/S0022-1236(03)00059-4 |
| [5] | Betancor, Jorge J., On Hardy spaces associated with Bessel operators, J. Anal. Math., 195-219 (2009) · Zbl 1195.44023 · doi:10.1007/s11854-009-0008-1 |
| [6] | Cao, Jun, Real interpolation of weighted tent spaces, Appl. Anal., 2415-2443 (2016) · Zbl 1362.46023 · doi:10.1080/00036811.2015.1091924 |
| [7] | Carron, Gilles, Riesz transform and \(L^p\)-cohomology for manifolds with Euclidean ends, Duke Math. J., 59-93 (2006) · Zbl 1106.58021 · doi:10.1215/S0012-7094-06-13313-6 |
| [8] | Chang, Der-Chen, \(H^p\) theory on a smooth domain in \(\mathbf{R}^N\) and elliptic boundary value problems, J. Funct. Anal., 286-347 (1993) · Zbl 0804.35027 · doi:10.1006/jfan.1993.1069 |
| [9] | Coifman, Ronald R., A real variable characterization of \(H^p\), Studia Math., 269-274 (1974) · Zbl 0289.46037 · doi:10.4064/sm-51-3-269-274 |
| [10] | Coifman, R. R., Some new function spaces and their applications to harmonic analysis, J. Funct. Anal., 304-335 (1985) · Zbl 0569.42016 · doi:10.1016/0022-1236(85)90007-2 |
| [11] | Coifman, Ronald R., Analyse harmonique non-commutative sur certains espaces homog\`enes, Lecture Notes in Mathematics, Vol. 242, v+160 pp. (1971), Springer-Verlag, Berlin-New York · Zbl 0224.43006 |
| [12] | Coifman, Ronald R., Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 569-645 (1977) · Zbl 0358.30023 · doi:10.1090/S0002-9904-1977-14325-5 |
| [13] | Duong, Xuan Thinh, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc., 943-973 (2005) · Zbl 1078.42013 · doi:10.1090/S0894-0347-05-00496-0 |
| [14] | Dziuba\'{n}ski, Jacek, On Riesz transforms characterization of \(H^1\) spaces associated with some Schr\"{o}dinger operators, Potential Anal., 39-50 (2011) · Zbl 1223.42009 · doi:10.1007/s11118-010-9202-0 |
| [15] | Dziuba\'{n}ski, Jacek, Hardy spaces for semigroups with Gaussian bounds, Ann. Mat. Pura Appl. (4), 965-987 (2018) · Zbl 1395.42053 · doi:10.1007/s10231-017-0711-y |
| [16] | Dziuba\'{n}ski, Jacek, Multivariate H\"{o}rmander-type multiplier theorem for the Hankel transform, J. Fourier Anal. Appl., 417-437 (2013) · Zbl 1311.42025 · doi:10.1007/s00041-013-9260-y |
| [17] | Dziuba\'{n}ski, Jacek, \(H^p\) spaces associated with Schr\"{o}dinger operators with potentials from reverse H\"{o}lder classes, Colloq. Math., 5-38 (2003) · Zbl 1083.42015 · doi:10.4064/cm98-1-2 |
| [18] | Dziuba\'{n}ski, Jacek, Hardy spaces \(H^1\) for Schr\"{o}dinger operators with certain potentials, Studia Math., 39-53 (2004) · Zbl 1061.47040 · doi:10.4064/sm164-1-3 |
| [19] | Dziuba\'{n}ski, Jacek, On Hardy spaces associated with certain Schr\"{o}dinger operators in dimension 2, Rev. Mat. Iberoam., 1035-1060 (2012) · Zbl 1364.42027 · doi:10.4171/RMI/702 |
| [20] | Dziuba\'{n}ski, Jacek, On isomorphisms of Hardy spaces associated with Schr\"{o}dinger operators, J. Fourier Anal. Appl., 447-456 (2013) · Zbl 1305.42025 · doi:10.1007/s00041-013-9262-9 |
| [21] | Dziuba\'{n}ski, Jacek, A characterization of Hardy spaces associated with certain Schr\"{o}dinger operators, Potential Anal., 917-930 (2014) · Zbl 1301.42039 · doi:10.1007/s11118-014-9400-2 |
| [22] | Fefferman, C., \(H^p\) spaces of several variables, Acta Math., 137-193 (1972) · Zbl 0257.46078 · doi:10.1007/BF02392215 |
| [23] | Garc\'{\i }a-Cuerva, Jos\'{e}, Weighted \(H^p\) spaces, Dissertationes Math. (Rozprawy Mat.), 63 pp. (1979) · Zbl 0434.42023 |
| [24] | Goldberg, David, A local version of real Hardy spaces, Duke Math. J., 27-42 (1979) · Zbl 0409.46060 |
| [25] | Grigor\cprime yan, Alexander, Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Differential Geom., 33-52 (1997) · Zbl 0865.58042 |
| [26] | Grigor’yan, Alexander, Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Mathematics, xviii+482 pp. (2009), American Mathematical Society, Providence, RI; International Press, Boston, MA · Zbl 1206.58008 · doi:10.1090/amsip/047 |
| [27] | Gyrya, Pavel, Neumann and Dirichlet heat kernels in inner uniform domains, Ast\'{e}risque, viii+144 pp. (2011) · Zbl 1222.58001 |
| [28] | Hofmann, Steve, Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, Mem. Amer. Math. Soc., vi+78 pp. (2011) · Zbl 1232.42018 · doi:10.1090/S0065-9266-2011-00624-6 |
| [29] | Hofmann, Steve, Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann., 37-116 (2009) · Zbl 1162.42012 · doi:10.1007/s00208-008-0295-3 |
| [30] | Hofmann, Steve, Second order elliptic operators with complex bounded measurable coefficients in \(L^p\), Sobolev and Hardy spaces, Ann. Sci. \'{E}c. Norm. Sup\'{e}r. (4), 723-800 (2011) · Zbl 1243.47072 · doi:10.24033/asens.2154 |
| [31] | Hu, Guorong, Littlewood-Paley characterization of weighted Hardy spaces associated with operators, J. Aust. Math. Soc., 250-267 (2017) · Zbl 1376.42023 · doi:10.1017/S1446788716000562 |
| [32] | Ishige, Kazuhiro, The heat kernel of a Schr\"{o}dinger operator with inverse square potential, Proc. Lond. Math. Soc. (3), 381-410 (2017) · Zbl 1379.35079 · doi:10.1112/plms.12041 |
| [33] | Kania-Strojec, Edyta, Local atomic decompositions for multidimensional Hardy spaces, Rev. Mat. Complut., 409-434 (2021) · Zbl 1481.42029 · doi:10.1007/s13163-020-00354-y |
| [34] | Kania-Strojec, Edyta, Riesz transform characterizations for multidimensional Hardy spaces, J. Geom. Anal., Paper No. 163, 37 pp. (2022) · Zbl 1485.42039 · doi:10.1007/s12220-022-00896-1 |
| [35] | Killip, R., Sobolev spaces adapted to the Schr\"{o}dinger operator with inverse-square potential, Math. Z., 1273-1298 (2018) · Zbl 1391.35123 · doi:10.1007/s00209-017-1934-8 |
| [36] | Latter, Robert H., A characterization of \(H^p(\mathbf{R}^n)\) in terms of atoms, Studia Math., 93-101 (1978) · Zbl 0398.42017 · doi:10.4064/sm-62-1-93-101 |
| [37] | Liskevich, Vitali, Estimates of integral kernels for semigroups associated with second-order elliptic operators with singular coefficients, Potential Anal., 359-390 (2003) · Zbl 1023.35041 · doi:10.1023/A:1021877025938 |
| [38] | Liu, Suying, An atomic decomposition of weighted Hardy spaces associated to self-adjoint operators, J. Funct. Anal., 2709-2723 (2013) · Zbl 1285.42019 · doi:10.1016/j.jfa.2013.08.003 |
| [39] | McIntosh, Alan, Miniconference on operator theory and partial differential equations. Operators which have an \(H_\infty\) functional calculus, Proc. Centre Math. Anal. Austral. Nat. Univ., 210-231 (1986), Austral. Nat. Univ., Canberra · Zbl 0634.47016 |
| [40] | Milman, Pierre D., Global heat kernel bounds via desingularizing weights, J. Funct. Anal., 373-398 (2004) · Zbl 1057.47043 · doi:10.1016/j.jfa.2003.12.008 |
| [41] | Preisner, Marcin, Riesz transform characterization of \(H^1\) spaces associated with certain Laguerre expansions, J. Approx. Theory, 229-252 (2012) · Zbl 1241.42017 · doi:10.1016/j.jat.2011.10.004 |
| [42] | Russ, Emmanuel, CMA/AMSI Research Symposium “Asymptotic Geometric Analysis, Harmonic Analysis, and Related Topics”. The atomic decomposition for tent spaces on spaces of homogeneous type, Proc. Centre Math. Appl. Austral. Nat. Univ., 125-135 (2007), Austral. Nat. Univ., Canberra · Zbl 1158.46020 |
| [43] | Sikora, Adam, Riesz transform, Gaussian bounds and the method of wave equation, Math. Z., 643-662 (2004) · Zbl 1066.58014 · doi:10.1007/s00209-003-0639-3 |
| [44] | Song, Liang, A \(q\)-atomic decomposition of weighted tent spaces on spaces of homogeneous type and its application, J. Geom. Anal., 3029-3059 (2021) · Zbl 1460.42038 · doi:10.1007/s12220-020-00382-6 |
| [45] | Song, Liang, A maximal function characterization for Hardy spaces associated to nonnegative self-adjoint operators satisfying Gaussian estimates, Adv. Math., 463-484 (2016) · Zbl 1334.42048 · doi:10.1016/j.aim.2015.09.026 |
| [46] | Song, Liang, Maximal function characterizations for Hardy spaces associated with nonnegative self-adjoint operators on spaces of homogeneous type, J. Evol. Equ., 221-243 (2018) · Zbl 1388.42064 · doi:10.1007/s00028-017-0398-y |
| [47] | Song, Renming, Estimates on the Dirichlet heat kernel of domains above the graphs of bounded \(C^{1,1}\) functions, Glas. Mat. Ser. III, 273-286 (2004) · Zbl 1064.35066 · doi:10.3336/gm.39.2.09 |
| [48] | Stein, Elias M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, xiv+695 pp. (1993), Princeton University Press, Princeton, NJ · Zbl 0821.42001 |
| [49] | Stein, Elias M., On the theory of harmonic functions of several variables. I. The theory of \(H^p\)-spaces, Acta Math., 25-62 (1960) · Zbl 0097.28501 · doi:10.1007/BF02546524 |
| [50] | Strauss, Walter A., Partial differential equations, xii+425 pp. (1992), John Wiley & Sons, Inc., New York · Zbl 0817.35001 |
| [51] | Str\"{o}mberg, Jan-Olov, Weighted Hardy spaces, Lecture Notes in Mathematics, vi+193 pp. (1989), Springer-Verlag, Berlin · Zbl 0676.42021 · doi:10.1007/BFb0091154 |
| [52] | Uchiyama, Akihito, A maximal function characterization of \(H^p\) on the space of homogeneous type, Trans. Amer. Math. Soc., 579-592 (1980) · Zbl 0503.46020 · doi:10.2307/1999848 |
| [53] | Yang, Dachun, Musielak-Orlicz-Hardy spaces associated with operators and their applications, J. Geom. Anal., 495-570 (2014) · Zbl 1302.42033 · doi:10.1007/s12220-012-9344-y |
| [54] | Zhang, Qi S., The global behavior of heat kernels in exterior domains, J. Funct. Anal., 160-176 (2003) · Zbl 1021.58019 · doi:10.1016/S0022-1236(02)00074-5 |
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