Endpoint boundedness of linear commutators on local Hardy spaces over metric measure spaces of homogeneous type. (English) Zbl 1462.42022
Summary: Let \((\mathcal{X},d,\mu)\) be a metric measure space of homogeneous type in the sense of R. R. Coifman and G. Weiss [Analyse harmonique non-commutative sur certains espaces homogènes. Etude de certaines intégrales singulières. (Non-commutative harmonic analysis on certain homogeneous spaces. Study of certain singular integrals.) Springer, Cham (1971; Zbl 0224.43006); Bull. Am. Math. Soc. 83, 569–645 (1977; Zbl 0358.30023)]. In this article, the authors prove that the commutator, generated by any \(b\in\text{BMO}(\mathcal{X})\) and any Calderón-Zygmund operator, is bounded from the Hardy type space \(H^1_b(\mathcal{X})\) to the local Hardy space \(H^1_{\rho}(\mathcal{X})\) associated with an admissible function \(\rho\), where \(H^1_b(\mathcal{X})\) is the largest subspace of the Hardy space \(H^1(\mathcal{X})\) that ensures the boundedness of commutators from \(H^1_b(\mathcal{X})\) to \(L^1(\mathcal{X})\). Moreover, the authors investigate the relations between the Hardy space \(H^1_L(\mathbb{R}^n)\) associated with the Schrödinger operator \(L\) and the local Hardy space \(h^1(\mathbb{R}^n)\). The major novelties of this article are that the main result even essentially improves the corresponding Euclidean case and, throughout this article, \(\mu\) is not assumed to satisfy the reverse doubling condition.
MSC:
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 42B30 | \(H^p\)-spaces |
| 47B47 | Commutators, derivations, elementary operators, etc. |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 30L99 | Analysis on metric spaces |
Keywords:
metric measure space of homogeneous type; commutator; local Hardy space; wavelet; bilinear decomposition; Calderón-Zygmund operatorReferences:
| [1] | Aalto, D.; Berkovits, L.; Kansanen, OE; Yue, H., John-Nirenberg lemmas for a doubling measure, Stud. Math., 204, 21-37 (2011) · Zbl 1230.43006 |
| [2] | Auscher, P.; Hytönen, T., Orthonormal bases of regular wavelets in spaces of homogeneous type, Appl. Comput. Harmon. Anal., 34, 266-296 (2013) · Zbl 1261.42057 |
| [3] | Bonami, A.; Iwaniec, T.; Jones, P.; Zinsmeister, M., On the product of functions in BMO and \(H^1\), Ann. Inst. Fourier (Grenoble), 57, 1405-1439 (2007) · Zbl 1132.42010 |
| [4] | Bonami, A.; Grellier, S.; Ky, LD, Paraproducts and products of functions in BMO \(({\mathbb{R}}^n)\) and \(H^1({\mathbb{R}}^n)\) through wavelets, J. Math. Pures Appl. (9), 97, 230-241 (2012) · Zbl 1241.47028 |
| [5] | Bui, TA; Duong, XT; Ky, LD, Hardy spaces associated to critical functions and applications to T1 theorems, J. Fourier Anal. Appl., 26, 67 (2020) · Zbl 1439.42032 |
| [6] | Chanillo, S., A note on commutators, Indiana Univ. Math. J., 31, 7-16 (1982) · Zbl 0523.42015 |
| [7] | Chen, Y.; Ding, Y.; Hong, G., Commutators with fractional differentiation and new characterizations of BMO-Sobolev spaces, Anal. PDE, 9, 1497-1522 (2016) · Zbl 1350.42022 |
| [8] | Coifman, R.R., Weiss, G.: Analyse Harmonique Non-commutative sur Certains Espaces Homogènes, (French) Étude de Certaines Intégrales Singulières. Lecture Notes in Mathematics, vol. 242. Springer-Verlag, Berlin-New York (1971) · Zbl 0224.43006 |
| [9] | Coifman, RR; Weiss, G., Extensions of Hardy spaces and their use in analysis, Bull. Am. Math. Soc., 83, 569-645 (1977) · Zbl 0358.30023 |
| [10] | Coifman, RR; Rochberg, R.; Weiss, G., Factorization theorems for Hardy spaces in several variables, Ann. Math. (2), 103, 611-635 (1976) · Zbl 0326.32011 |
| [11] | Deng, D., Han, Y.: Harmonic Analysis on Spaces of Homogeneous Type. Lecture Notes in Mathematics, vol. 1966. Springer-Verlag, Berlin (2009) · Zbl 1158.43002 |
| [12] | Dinh, TD; Hung, HD; Ky, LD, On weak \(*\)-convergence in the localized Hardy spaces \(H^1_\rho ({\cal{X}})\) and its application, Taiwan. J. Math., 20, 897-907 (2016) · Zbl 1357.42022 |
| [13] | Duoandikoetxea, J.: Fourier Analysis,Translated and revised from the 1995 Spanish original by David Cruz-Uribe, Graduate Studies in Mathematics 29, American Mathematical Society, Providence, RI, (2001) · Zbl 0969.42001 |
| [14] | Duong, XT; Li, J.; Wick, B.; Yang, D., Factorization for Hardy spaces and characterization for BMO spaces via commutators in the Bessel setting, Indiana Univ. Math. J., 66, 1081-1106 (2017) · Zbl 1376.42028 |
| [15] | Dziubański, J.; Zienkiewicz, J., Hardy space \(H^1\) associated to Schrödinger operator with potential satisfying reverse Hölder inequality, Rev. Mat. Iberoam., 15, 279-296 (1999) · Zbl 0959.47028 |
| [16] | Dziubański, J.; Zienkiewicz, J., Hardy spaces \(H^1\) for Schrödinger operators with compactly supported potentials, Ann. Mat. Pura Appl. (4), 184, 315-326 (2005) · Zbl 1223.42014 |
| [17] | Ferguson, SH; Lacey, MT, A characterization of product BMO by commutators, Acta Math., 189, 143-160 (2002) · Zbl 1039.47022 |
| [18] | Folland, GB, Real Analysis. Modern Techniques and Their Applications, Second Edition, Pure and Applied Mathematics (New York) (1999), New York: Wiley, New York · Zbl 0924.28001 |
| [19] | Fu, X.; Yang, D., Products of functions in \(H^1_{\rho } ({\cal{X}})\) and \({{\rm BMO}}_{\rho } (X)\) over RD-spaces and applications to Schrödinger operators, J. Geom. Anal., 27, 2938-2976 (2017) · Zbl 1388.42061 |
| [20] | Fu, X.; Yang, D., Wavelet characterizations of the atomic Hardy space \(H^1\) on spaces of homogeneous type, Appl. Comput. Harmon. Anal., 44, 1-37 (2018) · Zbl 1381.42032 |
| [21] | Fu, X.; Chang, D-C; Yang, D., Recent progress in bilinear decompositions, Appl. Anal. Optim., 1, 153-210 (2017) · Zbl 1483.42013 |
| [22] | Fu, X.; Yang, D.; Liang, Y., Products of functions in BMO \(({{\cal{X}}})\) and \(H^1_{{\rm at}}({{\cal{X}}})\) via wavelets over spaces of homogeneous type, J. Fourier Anal. Appl., 23, 919-990 (2017) · Zbl 1426.42018 |
| [23] | Fu, X.; Ma, T.; Yang, D., Real-variable characterizations of Musielak-Orlicz Hardy spaces on spaces of homogeneous type, Ann. Acad. Sci. Fenn. Math., 45, 343-410 (2020) · Zbl 1437.42027 |
| [24] | García-Cuerva, J.; Rubio de Francia, JL, Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies 116 (1985), Amsterdam: North-Holland Publishing Co., Amsterdam · Zbl 0578.46046 |
| [25] | Gehring, FW, The \(L^p\)-integrability of the partial derivatives of a quasiconformal mapping, Acta Math., 130, 265-277 (1973) · Zbl 0258.30021 |
| [26] | Goldberg, D., A local version of real Hardy spaces, Duke Math. J., 46, 27-42 (1979) · Zbl 0409.46060 |
| [27] | Grafakos, L., Hardy space estimates for multilinear operators II, Rev. Mater. Iberoam., 8, 69-92 (1992) · Zbl 0785.47026 |
| [28] | Grafakos, L., Modern Fourier Analysis, Third edition, Graduate Texts in Mathematics 250 (2014), New York: Springer, New York · Zbl 1304.42002 |
| [29] | Grafakos, L., Classical Fourier Analysis, Third edition, Graduate Texts in Mathematics 249 (2014), New York: Springer, New York · Zbl 1304.42001 |
| [30] | Han, Y.; Müller, D.; Yang, D., Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type, Math. Nachr., 279, 1505-1537 (2006) · Zbl 1179.42016 |
| [31] | Han, Y.; Müller, D.; Yang, D., A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces, Abstr. Appl. Anal., 893409, 250 (2008) · Zbl 1193.46018 |
| [32] | He, Z.; Han, Y.; Li, J.; Liu, L.; Yang, D.; Yuan, W., A complete real-variable theory of Hardy spaces on spaces of homogeneous type, J. Fourier Anal. Appl., 25, 2197-2267 (2019) · Zbl 1427.42026 |
| [33] | He, Z.; Liu, L.; Yang, D.; Yuan, W., New Calderón reproducing formulae with exponential decay on spaces of homogeneous type, Sci. China Math., 62, 283-350 (2019) · Zbl 1409.42027 |
| [34] | He, Z.; Yang, D.; Yuan, W., Real-variable characterizations of local Hardy spaces on spaces of homogeneous type, Math. Nachr. (2019) · Zbl 1528.42026 · doi:10.1002/mana.201900320 |
| [35] | Heinonen, J., Lectures on Analysis on Metric Spaces (2001), New York: Universitext, Springer, New York · Zbl 0985.46008 |
| [36] | Hu, G.; Yang, D.; Zhou, Y., Boundedness of singular integrals in Hardy spaces on spaces of homogeneous type, Taiwan. J. Math., 13, 91-135 (2009) · Zbl 1177.42011 |
| [37] | Hytönen, T.; Kairema, A., Systems of dyadic cubes in a doubling metric space, Colloq. Math., 126, 1-33 (2012) · Zbl 1244.42010 |
| [38] | Hytönen, T.; Tapiola, O., Almost Lipschitz-continuous wavelets in metric spaces via a new randomization of dyadic cubes, J. Approx. Theory, 185, 12-30 (2014) · Zbl 1302.42054 |
| [39] | Killip, R.; Miao, C.; Visan, M.; Zhang, J.; Zheng, J., Sobolev spaces adapted to the Schrödinger operator with inverse-square potential, Math. Z., 288, 1273-1298 (2018) · Zbl 1391.35123 |
| [40] | Kurata, K., An estimate on the heat kernel of magnetic Schrödinger operators and uniformly elliptic operators with non-negative potentials, J. Lond. Math. Soc. (2), 62, 885-903 (2000) · Zbl 1013.35020 |
| [41] | Ky, LD, Bilinear decompositions and commutators of singular integral operators, Trans. Am. Math. Soc., 365, 2931-2958 (2013) · Zbl 1272.42010 |
| [42] | Ky, LD, Bilinear decompositions for the product space \(H^1_L\times{{\rm BMO}}_L\), Math. Nachr., 287, 1288-1297 (2014) · Zbl 1305.42027 |
| [43] | Ky, LD, On the product of functions in BMO and \(H^1\) over spaces of homogeneous type, J. Math. Anal. Appl., 425, 807-817 (2015) · Zbl 1334.46024 |
| [44] | Lin, H.; Wu, S.; Yang, D., Boundedness of certain commutators over non-homogeneous metric measure spaces, Anal. Math. Phys., 7, 187-218 (2017) · Zbl 1367.47043 |
| [45] | Liu, L.; Chang, D-C; Fu, X.; Yang, D., Endpoint boundedness of commutators on spaces of homogeneous type, Appl. Anal., 96, 2408-2433 (2017) · Zbl 06816838 |
| [46] | Liu, L.; Chang, D-C; Fu, X.; Yang, D., Endpoint estimates of linear commutators on Hardy spaces over spaces of homogeneous type, Math. Method Appl. Sci., 41, 5951-5984 (2018) · Zbl 1403.42012 |
| [47] | Macías, RA; Segovia, C., A decomposition into atoms of distributions on spaces of homogeneous type, Adv. Math., 33, 271-309 (1979) · Zbl 0431.46019 |
| [48] | Muckenhoupt, B., Weighted norm inequalities for the Hardy maximal function, Trans. Am. Math. Soc., 165, 207-226 (1972) · Zbl 0236.26016 |
| [49] | Nakai, E.; Yabuta, K., Pointwise multipliers for functions of weighted bounded mean oscillation on spaces of homogeneous type, Math. Japon., 46, 15-28 (1997) · Zbl 0884.42010 |
| [50] | Pérez, C., Endpoint estimates for commutators of singular integral operators, J. Funct. Anal., 128, 163-185 (1995) · Zbl 0831.42010 |
| [51] | Shen, Z., \(L^p\) estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble), 45, 513-546 (1995) · Zbl 0818.35021 |
| [52] | Wojtaszczyk, P., Banach Spaces for Analysts, Cambridge Studies in Advanced Mathematics 25 (1991), Cambridge: Cambridge University Press, Cambridge · Zbl 0724.46012 |
| [53] | Yang, D.; Zhou, Y., Localized Hardy spaces \(H^1\) related to admissible functions on RD-spaces and applications to Schrödinger operators, Trans. Am. Math. Soc., 363, 1197-1239 (2011) · Zbl 1217.42044 |
| [54] | Yang, D.; Zhou, Y., New properties of Besov and Triebel-Lizorkin spaces on RD-spaces, Manuscr. Math., 134, 59-90 (2011) · Zbl 1215.46026 |
| [55] | Yang, D.; Yang, D.; Zhou, Y., Localized BMO and BLO spaces on RD-spaces and applications to Schrödinger operators, Commun. Pure Appl. Anal., 9, 779-812 (2010) · Zbl 1188.42008 |
| [56] | Yang, D.; Yang, D.; Zhou, Y., Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrödinger operators, Nagoya Math. J., 198, 77-119 (2010) · Zbl 1214.46019 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
