New Molecular Characterization of Musielak-Orlicz Hardy Spaces on Spaces of Homogeneous Type and Its Applications

Abstract

Let

(\mathcal{X},d,\mu )

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(\cal{X},d,\mu)$$\end{document}

be a space of homogeneous type, in the sense of Coifman and Weiss, and

\phi :\mathcal{X}\times [0,\mathrm{\infty })\to [0,\mathrm{\infty })

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varphi:\cal{X}\times[0,\infty)\rightarrow[0,\infty)$$\end{document}

satisfy that, for almost every

x\in \mathcal{X},\phi (x,\cdot )

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x\in\cal{X},\varphi(x,\cdot)$$\end{document}

is an Orlicz function and that φ(·, t) is a Muckenhoupt

{\mathbb{A}}_{\mathrm{\infty }}(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{A}_{\infty}(\cal{X})$$\end{document}

weight uniformly in t ∈ [0, ∞). In this article, the authors first establish a new molecular characterization, associated with admissible sequences of balls on

\mathcal{X}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\cal{X}$$\end{document}

, of the Musielak-Orlicz Hardy space

H^{\phi }(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^{\varphi}(\cal{X})$$\end{document}

. As an application, the authors also obtain the boundedness of Calderón-Zygmund operators from

H^{\phi }(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^{\varphi}(\cal{X})$$\end{document}

to

H^{\phi }(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^{\varphi}(\cal{X})$$\end{document}

or to the Musielak-Orlicz space

L^{\phi }(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^{\varphi}(\cal{X})$$\end{document}

. The main novelty of these results is that, in the proof of the boundedness of Calderón-Zygmund operators on

H^{\phi }(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^{\varphi}(\cal{X})$$\end{document}

, the authors get rid of the dependence on the reverse doubling property of μ by using this new molecular characterization of

H^{\phi }(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^{\varphi}(\cal{X})$$\end{document}

.

Keywords

Space of homogeneous type / Musielak-Orlicz function / Hardy space / Molecule / Calderón-Zygmund operator

References

[1] Abu-ShammalaW, TorchinskyA. The Hardy-Lorentz spaces H^p,q(ℝⁿ). Studia Math., 2007, 182: 283-294

[2] AuscherP, HytönenT. Orthonormal bases of regular wavelets in spaces of homogeneous type. Appl. Comput. Harmon. Anal., 2013, 34: 266-296

[3] BonamiA, CaoJ, KyL D, et al.. Multiplication between Hardy spaces and their dual spaces. J. Math. Pures Appl. (9), 2019, 131: 130-170

[4] BonamiA, FeutoJ, GrellierS. Endpoint for the DIV-CURL lemma in Hardy spaces. Publ. Mat., 2010, 54: 341-358

[5] BonamiA, GrellierS, KyL D. Paraproducts and products of functions in BMO(ℝⁿ) and H¹(ℝⁿ) through wavelets. J. Math. Pures Appl. (9), 2012, 97: 230-241

[6] BonamiA, IwaniecT, JonesP, ZinsmeisterM. On the product of functions in BMO and H¹. Ann. Inst. Fourier (Grenoble), 2007, 57: 1405-1439

[7] Bonami, A., Ky, L. D., Liang, Y. and Yang, D., Several remarks on Musielak-Orlicz-Hardy spaces, Bull. Sci. Math., 181, 2022, Paper No. 103206, 35 pp.

[8] BonamiA, LiuL, YangD, YuanW. Pointwise multipliers of Zygmund classes on ℝⁿ. J. Geom. Anal., 2021, 31: 8879-8902

[9] BuiT A, CaoJ, KyL D, et al.. Musielak-Orlicz-Hardy spaces associated with operators satisfying reinforced off-diagonal estimates. Anal. Geom. Metr. Spaces, 2013, 1: 69-129

[10] BuiT A, LyF K, YangS. Second-order Riesz transforms associated with magnetic Schrödinger operators. J. Math. Anal. Appl., 2016, 437: 1196-1218

[11] CaoJ, ChangD-C, YangD, YangS. Riesz transform characterizations of Musielak-Orlicz Hardy spaces. Trans. Amer. Math. Soc., 2016, 368: 6979-7018

[12] Cao, J., Ky, L. D. and Yang, D., Bilinear decompositions of products of local Hardy and Lipschitz or BMO spaces through wavelets, Commun. Contemp. Math., 20 (3), 2018, Paper No. 1750025, 30 pp.

[13] ChangD-C, FuZ, YangD, YangS. Real-variable characterizations of Musielak-Orlicz Hardy spaces associated with Schrödinger operators on domains. Math. Methods Appl. Sci., 2016, 39: 533-569

[14] CleanthousG, GeorgiadisA G, NielsenM. Anisotropic mixed-norm Hardy spaces. J. Geom. Anal., 2017, 27: 2758-2787

[15] CoifmanR R, LionsP-L, MeyerY, SemmesS. Compensated compactness and Hardy spaces. J. Math. Pures Appl. (9), 1993, 72: 247-286

[16] CoifmanR R, WeissG. FrAnalyse Harmonique Non-Commutative sur Certains Espaces Homogènes. Étude de Certaines Intégrales Singulières, 1971, Berlin, New York, Springer-Verlag(French)242

[17] CoifmanR R, WeissG. Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc., 1977, 83: 569-645

[18] DengD, HanYHarmonic Analysis on Spaces of Homogeneous Type, 2009, Berlin, Springer-Verlag1966

[19] DuongX-T, TranT D. Musielak-Orlicz Hardy spaces associated to operators satisfying Davies–Gaffney estimates and bounded holomorphic functional calculus. J. Math. Soc. Japan, 2016, 68: 1-30

[20] DuongX-T, YanL. Hardy spaces of spaces of homogeneous type. Proc. Amer. Math. Soc., 2003, 131: 3181-3189

[21] FeffermanC, SteinE M. H^p spaces of several variables. Acta Math., 1972, 129: 137-193

[22] FuX, MaT, YangD. Real-variable characterizations of Musielak-Orlicz Hardy spaces on spaces of homogeneous type. Ann. Acad. Sci. Fenn. Math., 2020, 45: 343-410

[23] FuX, YangD. Wavelet characterizations of Musielak-Orlicz Hardy spaces. Banach J. Math. Anal., 2018, 12: 1017-1046

[24] GrafakosL, LiuL, YangD. Maximal function characterizations of Hardy spaces on RD-spaces and their applications. Sci. China Ser. A, 2008, 51: 2253-2284

[25] HanY, LiJ, WardL A. Hardy space theory on spaces of homogeneous type via orthonormal wavelet bases. Appl. Comput. Harmon. Anal., 2018, 45: 120-169

[26] HanY, MüllerD, YangD. Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type. Math. Nachr., 2006, 279: 1505-1537

[27] Han, Y., Müller, D. and Yang, D., A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces, Abstr. Appl. Anal., 2008, 2008, Article ID 893409, 250 pp.

[28] HanY, SawyerE T. Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces. Mem. Amer. Math. Soc., 1994, 110(530): 1-126

[29] HeZ, HanY, LiJ, et al.. A complete real-variable theory of Hardy spaces on spaces of homogeneous type. J. Fourier Anal. Appl., 2019, 25: 2197-2267

[30] HeZ, LiuL, YangD, YuanW. New Calderón reproducing formulae with exponential decay on spaces of homogeneous type. Sci. China Math., 2019, 62: 283-350

[31] HeZ, YangD, YuanW. Real-variable characterizations of local Hardy spaces on spaces of homogeneous type. Math. Nachr., 2021, 294: 900-955

[32] HoK-P. Atomic decomposition of Hardy-Morrey spaces with variable exponents. Ann. Acad. Sci. Fenn. Math., 2015, 40: 31-62

[33] HoK-P. Sublinear operators on weighted Hardy spaces with variable exponents. Forum Math., 2019, 31: 607-617

[34] HouS, YangD, YangS. Lusin area function and molecular characterizations of Musielak-Orlicz Hardy spaces and their applications. Commun. Contemp. Math., 2013, 15: 1350029 37 pp

[35] HouS, YangD, YangS. Musielak-Orlicz BMO-type spaces associated with generalized approximations to the identity. Acta Math. Sin. (Engl. Ser.), 2014, 30: 1917-1962

[36] HuyD Q, KyL D. Boundedness of fractional integral operators on Musielak-Orlicz Hardy spaces. Math. Nachr., 2021, 294: 2340-2354

[37] HytönenT, KairemaA. Systems of dyadic cubes in a doubling metric space. Colloq. Math., 2012, 126: 1-33

[38] IwaniecT, OnninenJ. H1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\cal{H}^{1}$$\end{document}-estimates of Jacobians by subdeterminants. Math. Ann., 2002, 324: 341-358

[39] Izuki, M., Nakai, E. and Sawano, Y., Atomic and wavelet characterization of Musielak-Orlicz Hardy spaces for generalized Orlicz functions, Integral Equations Operator Theory, 94, 2022, Paper No. 3, 33 pp.

[40] JansonS. Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation. Duke Math. J., 1980, 47: 959-982

[41] JohnF, NirenbergL. On functions of bounded mean oscillation. Comm. Pure Appl. Math., 1961, 14: 415-426

[42] KyL D. Bilinear decompositions and commutators of singular integral operators. Trans. Amer. Math. Soc., 2013, 365: 2931-2958

[43] KyL D. New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators. Integral Equations Operator Theory, 2014, 78: 115-150

[44] KyL D. Endpoint estimates for commutators of singular integrals related to Schrödinger operators. Rev. Mat. Iberoam., 2015, 31: 1333-1373

[45] LiB, FanX, FuZ, YangD. Molecular characterization of anisotropic Musielak-Orlicz Hardy spaces and their applications. Acta Math. Sin. (Engl. Ser.), 2016, 32: 1391-1414

[46] Li, B., Yang, D. and Yuan, W., Anisotropic Hardy spaces of Musielak-Orlicz type with applications to boundedness of sublinear operators, The Scientific World Journal, 2014, 2014, Article ID 306214, 19 pp.

[47] LiJ, WardL A. Singular integrals on Carleson measure spaces CMO^p on product spaces of homogeneous type. Proc. Amer. Math. Soc., 2013, 141: 2767-2782

[48] LiangY, NakaiE, YangD, ZhangJ. Boundedness of intrinsic Littlewood-Paley functions on Musielak-Orlicz Morrey and Campanato spaces. Banach J. Math. Anal., 2014, 8: 221-268

[49] LiangY, YangD. Musielak-Orlicz Campanato spaces and applications. J. Math. Anal. Appl., 2013, 406: 307-322

[50] LiangY, YangD. Intrinsic square function characterizations of Musielak-Orlicz Hardy spaces. Trans. Amer. Math. Soc., 2015, 367: 3225-3256

[51] LiangY, YangD, JiangR. Weak Musielak-Orlicz Hardy spaces and applications. Math. Nachr., 2016, 289: 634-677

[52] LiuJ, HaroskeD D, YangD. New molecular characterizations of anisotropic Musielak-Orlicz Hardy spaces and their applications. J. Math. Anal. Appl., 2019, 475: 1341-1366

[53] LiuJ, HaroskeD D, YangD, YuanW. Dual spaces and wavelet characterizations of anisotropic Musielak-Orlicz Hardy spaces. Appl. Comput. Math., 2020, 19: 106-131

[54] LiuL, ChangD-C, FuX, YangD. Endpoint estimates of linear commutators on Hardy spaces over spaces of homogeneous type. Math. Methods Appl. Sci., 2018, 41: 5951-5984

[55] MacíasR A, SegoviaC. A decomposition into atoms of distributions on spaces of homogeneous type. Adv. Math., 1979, 33: 271-309

[56] MüllerS. Hardy space methods for nonlinear partial differential equations. Tatra Mt. Math. Publ., 1994, 4: 159-168

[57] NakaiE, SawanoY. Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal., 2012, 262: 3665-3748

[58] NakaiE, YabutaK. Pointwise multipliers for functions of weighted bounded mean oscillation on spaces of homogeneous type. Math. Japon., 1997, 46: 15-28

[59] SawanoY, HoK-P, YangD, YangS. Hardy spaces for ball quasi-Banach function spaces. Dissertationes Math., 2017, 525: 1-102

[60] SawanoY, ShimomuraT, TanakaH. A remark on modified Morrey spaces on metric measure spaces. Hokkaido Math. J., 2018, 47: 1-15

[61] SteinE MHarmonic Analysis: Real-Variable Methods, Orthogonality, and Osicllatory Integrals, 1993, Princeton, NJ, Princeton University Press43

[62] SteinE M, WeissG. On the theory of harmonic functions of several variables, I, The theory of H^p-spaces. Acta Math., 1960, 103: 25-62

[63] StrömbergJ-O, TorchinskyAWeighted Hardy Spaces, 1989, Berlin, Springer-Verlag1381

[64] TaiblesonM H, WeissG. The molecular characterization of certain Hardy spaces, Representation theorems for Hardy spaces. Astérisque, 1980, 77: 67-149

[65] TranT D. Musielak-Orlicz Hardy spaces associated with divergence form elliptic operators without weight assumptions. Nagoya Math. J., 2014, 216: 71-110

[66] YanX, HeZ, YangD, YuanW. Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type: Littlewood-Paley characterizations with applications to boundedness of Calderón-Zygmund operators. Acta Math. Sin. (Engl. Ser.), 2022, 38: 1133-1184

[67] YanX, HeZ, YangD, YuanW. Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces. Math. Nachr., 2023, 296: 3056-3116

[68] YanX, YangD, YuanW. Intrinsic square function characterizations of Hardy spaces associated with ball quasi-Banach function spaces. Front. Math. China, 2020, 15: 769-806

[69] YangD, LiangY, KyL DReal-Variable Theory of Musielak-Orlicz Hardy Spaces, 2017, Cham, Springer-Verlag2182

[70] YangD, YangS. Atomic and maximal function characterizations of Musielak-Orlicz-Hardy spaces associated to non-negative self-adjoint operators on spaces of homogeneous type. Collect. Math., 2019, 70: 197-246

[71] Yang, D., Yuan, W. and Zhang, Y., Bilinear decomposition and divergence-curl estimates on products related to local Hardy spaces and their dual spaces, J. Funct. Anal., 280, 2021, Paper No. 108796, 74 pp.

[72] YangD, ZhouY. Radial maximal function characterizations of Hardy spaces on RD-spaces and their applications. Math. Ann., 2010, 346: 307-333

[73] Zhang, Y., Yang, D. and Yuan, W., Real-variable characterizations of local Orlicz-slice Hardy spaces with application to bilinear decompositions, Commun. Contemp. Math., 24, 2022, Paper No. 2150004, 35 pp.

[74] ZhouX, HeZ, YangD. Real-variable characterizations of Hardy-Lorentz spaces on spaces of homogeneous type with applications to real interpolation and boundedness of Calderón-Zygmund operators. Anal. Geom. Metr. Spaces, 2020, 8: 182-260

RIGHTS & PERMISSIONS

The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg