Hardy Spaces Associated with Non-negative Self-adjoint Operators and Ball Quasi-Banach Function Spaces on Doubling Metric Measure Spaces and Their 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}$$({\mathcal {X}},d,\mu )$$\end{document}

be a doubling metric measure space in the sense of R. R. Coifman and G. Weiss, L a non-negative self-adjoint operator on

L^2(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^2({\mathcal {X}})$$\end{document}

satisfying the Davies–Gaffney estimate, and

X(\mathcal{X})

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

a ball quasi-Banach function space 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}$${\mathcal {X}}$$\end{document}

satisfying some extra mild assumptions. In this article, the authors introduce the Hardy type space

H_{X,L}(\mathcal{X})

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

by the Lusin area function associated with L and establish the atomic and the molecular characterizations of

H_{X,L}(\mathcal{X}).

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

As an application of these characterizations of

H_{X,L}(\mathcal{X})

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

, the authors obtain the boundedness of spectral multiplies on

H_{X,L}(\mathcal{X})

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

. Moreover, when L satisfies the Gaussian upper bound estimate, the authors further characterize

H_{X,L}(\mathcal{X})

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

in terms of the Littlewood–Paley functions

g_L

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

and

g_{\lambda ,L}^{\ast }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$g_{\lambda ,\,L}^*$$\end{document}

and establish the boundedness estimate of Schrödinger groups on

H_{X,L}(\mathcal{X})

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

. Specific spaces

X(\mathcal{X})

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

to which these results can be applied include Lebesgue spaces, Orlicz spaces, weighted Lebesgue spaces, and variable Lebesgue spaces. This shows that the results obtained in the article have extensive generality.

Keywords

Hardy space / Ball quasi-Banach function space / Non-negative self-adjoint operator / Atom / Molecule / Schrölder group / Spectral multiplier / Littlewood–Paley function / 42B30 / 42B25 / 35K08 / 42B35 / 35J30

References

[1] Amenta, A.: Tent spaces over metric measure spaces under doubling and related assumptions. In: Operator Theory in Harmonic and Non-commutative Analysis: 23rd International Workshop in Operator Theory and its Applications, vol. 240, pp. 1–29. Birkhäuser/Springer, Cham (2014)

[2] Auscher, P., Duong, X.T., McIntosh, A.: Boundedness of Banach space valued singular integral operators and Hardy spaces, Unpublished manuscript (2005)

[3] Auscher P, Hytönen T. Orthonormal bases of regular wavelets in spaces of homogeneous type. Appl. Comput. Harmon. Anal., 2013, 34: 266-296

[4] Bennett C, Sharpley R. Interpolation of Operators, Pure and Applied Mathematics, 1988, Boston, Academic Press Inc.129

[5] Bui TA. Weighted Hardy spaces associated to discrete Laplacians on graphs and applications. Potential Anal., 2014, 41: 817-848

[6] Bui TA, Cao J, Ky LD, Yang D, Yang S. Musielak–Orlicz–Hardy spaces associated with operators satisfying reinforced off-diagonal estimates. Anal. Geom. Metr. Sp., 2013, 1: 69-129

[7] Bui TA, Cao J, Ky LD, Yang D, Yang S. Weighted Hardy spaces associated with operators satisfying reinforced off-diagonal estimates. Taiwan. J. Math., 2013, 17: 1127-1166

[8] Bui TA, D’Ancona P, Duong XT, Müller D. On the flows associated to selfadjoint operators on metric measure spaces. Math. Ann., 2019, 375: 1393-1426

[9] Bui TA, D’Ancona P, Nicola F. Sharp Lp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document} estimates for Schrödinger groups on spaces of homogeneous type. Rev. Mat. Iberoam., 2020, 36: 455-484

[10] Bui TA, Li J. Orlicz–Hardy spaces associated to operators satisfying bounded H∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^\infty $$\end{document} functional calculus and Davies–Gaffney estimates. J. Math. Anal. Appl., 2011, 373: 485-501

[11] Bui TA, Ly FK. Sharp estimates for Schrödinger groups on Hardy spaces for 0<p≤1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0<p\le 1$$\end{document}. J. Fourier Anal. Appl., 2022, 28: 70

[12] Calderón A-P. Cauchy integrals on Lipschitz curves and related operators. Proc. Nat. Acad. Sci. U.S.A., 1977, 74: 1324-1327

[13] Calderón A-P, Calderón CP, Fabes E, Jodeit M, Rivière NM. Applications of the Cauchy integral on Lipschitz curves. Bull. Am. Math. Soc., 1978, 84: 287-290

[14] Chen P, Duong XT, Li J, Yan L. Sharp endpoint Lp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document} estimates for Schrödinger groups. Math. Ann., 2020, 378: 667-702

[15] Chen P, Duong XT, Li J, Yan L. Sharp endpoint estimates for Schrödinger groups on Hardy spaces. J. Differ. Equ., 2023, 371: 660-690

[16] Coifman RR, Meyer Y, Stein EM. Some new function spaces and their applications to harmonic analysis. J. Funct. Anal., 1985, 62: 304-335

[17] Coifman RR, Weiss G. Analyse Harmonique Non-commutative sur Certains Espaces Homogènes. (French) Étude de Certaines Intégrales Singulières, 1971, Berlin-New York, Springer-Verlag 242

[18] Coifman RR, Weiss G. Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc., 1977, 83: 569-645

[19] Coulhon T, Sikora A. Gaussian heat kernel upper bounds via the Phragmen–Lindelof theorem. Proc. Lond. Math. Soc. (3), 2008, 96: 507-544

[20] Dai F, Grafakos L, Pan Z, Yang D, Yuan W, Zhang Y. The Bourgain–Brezis–Mironescu formula on ball Banach function spaces. Math. Ann., 2023

[21] Dai F, Lin X, Yang D, Yuan W, Zhang Y. Brezis–Van Schaftingen–Yung formulae in ball Banach function spaces with applications to fractional Sobolev and Gagliardo–Nirenberg inequalities. Calc. Var. Partial Differ. Equ., 2023, 62: 56

[22] Davies EB. Heat kernel bounds, conservation of probability and the Feller property. J. Anal. Math., 1992, 58: 99-119

[23] del Campo R, Fernández A, Mayoral F, Naranjo F. Orlicz spaces associated to a quasi-Banach function space: applications to vector measures and interpolation. Collect. Math., 2021, 72: 481-499

[24] Duong XT, Yan L. Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J. Am. Math. Soc., 2005, 18: 943-973

[25] Duong XT, Yan L. New function spaces of BMO type, the John–Nirenberg inequality, interpolation, and applications. Commun. Pure Appl. Math., 2005, 58: 1375-1420

[26] Duong XT, Yan L. Spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies–Gaffney estimates. J. Math. Soc. Jpn., 2011, 63: 295-319

[27] Fefferman C, Stein EM. Hp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^p$$\end{document} spaces of several variables. Acta Math., 1972, 129: 137-193

[28] Fu X, Ma T, Yang D. Real-variable characterizations of Musielak–Orlicz Hardy spaces on spaces of homogeneous type. Ann. Acad. Sci. Fenn. Math., 2020, 45: 343-410

[29] Georgiadis AG, Kerkyacharian G, Kyriazis G, Petrushev P. Homogeneous Besov and Triebel–Lizorkin spaces associated to non-negative self-adjoint operators. J. Math. Anal. Appl., 2017, 449: 1382-1412

[30] Georgiadis AG, Kerkyacharian G, Kyriazis G, Petrushev P. Atomic and molecular decomposition of homogeneous spaces of distributions associated to non-negative self-adjoint operators. J. Fourier Anal. Appl., 2019, 25: 3259-3309

[31] Georgiadis AG, Kyriazis G. Embeddings between Triebel–Lizorkin spaces on metric spaces associated with operators. Anal. Geom. Metr. Sp., 2020, 8: 418-429

[32] Georgiadis AG, Kyriazis G. Duals of Besov and Triebel–Lizorkin spaces associated with operators. Constr. Approx., 2023, 57: 547-577

[33] Georgiadis AG, Nielsen M. Pseudodifferential operators on spaces of distributions associated with non-negative self-adjoint operators. J. Fourier Anal. Appl., 2017, 23: 344-378

[34] Georgiadis AG, Nielsen M. Spectral multipliers on spaces of distributions associated with non-negative self-adjoint operators. J. Approx. Theory, 2018, 234: 1-19

[35] Gong R, Yan L. Littlewood–Paley and spectral multipliers on weighted Lp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document} spaces. J. Geom. Anal., 2014, 24: 873-900

[36] Grafakos L, Liu L, Yang D. Vector-valued singular integrals and maximal functions on spaces of homogeneous type. Math. Scand., 2009, 104: 296-310

[37] Heinonen J. Lectures on Analysis on Metric Spaces, 2001, New York, Universitext, Springer-Verlag

[38] Hofmann S , Lu G, Mitrea D, Mitrea M, Yan L. Hardy spaces associated to non-negative self-adjoint operators satisfying Davies–Gaffney estimates. Mem. Am. Math. Soc., 2011, 214(1007): vi+78

[39] Hofmann S, Mayboroda S, McIntosh A. Second order elliptic operators with complex bounded measurable coefficients in Lp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document}, Sobolev and Hardy spaces. Ann. Sci. École Norm. Sup. (4), 2011, 44: 723-800

[40] Hu G. Littlewood–Paley characterization of weighted Hardy spaces associated with operators. J. Aust. Math. Soc., 2017, 103: 250-267

[41] Jiang R, Yang D. New Orlicz–Hardy spaces associated with divergence form elliptic operators. J. Funct. Anal., 2010, 258: 1167-1224

[42] Jiang R, Yang D. Orlicz–Hardy spaces associated with operators satisfying Davies–Gaffney estimates. Commun. Contemp. Math., 2011, 13: 331-373

[43] Kenig CE. Weighted Hp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^p$$\end{document} spaces on Lipschitz domains. Am. J. Math., 1980, 102: 129-163

[44] Lindenstrauss J, Tzafriri L. Classical Banach Spaces. II: Function Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, 1979, Berlin-New York, Springer-Verlag

[45] Liu J, Yang D, Zhang M. Sharp bilinear decomposition for products of both anisotropic Hardy spaces and their dual spaces with its applications to endpoint boundedness of commutators. Sci. China Math., 2023

[46] Liu S, Song L. An atomic decomposition of weighted Hardy spaces associated to self-adjoint operators. J. Funct. Anal., 2013, 265: 2709-2723

[47] McIntosh, A.: Operators which have an H∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H_\infty $$\end{document} functional calculus. In: Miniconference on Operator Theory and Partial Differential Equations (North Ryde, 1986) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 14, pp. 210–231. Australian National University, Canberra (1986)

[48] Nakai E, Yabuta K. Pointwise multipliers for functions of weighted bounded mean oscillation on spaces of homogeneous type. Math. Jpn., 1997, 46: 15-28

[49] Pan, Z., Yang, D., Yuan, W., Zhang, Y.: Gagliardo representation of norms of ball quasi-Banach function spaces. J. Funct. Anal. 286, 110205 (2024)

[50] Rao MM, Ren ZD. Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 1991, New York, Marcel Dekker Inc.

[51] Russ, E.: The atomic decomposition for tent spaces on spaces of homogeneous type. In: CMA/AMSI Research Symposium “Asymptotic Geometric Analysis, Harmonic Analysis, and Related Topics, Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 42, pp. 125–135. Australian National University, Canberra (2007)

[52] Sawano Y, Ho K-P, Yang D, Yang S. Hardy spaces for ball quasi-Banach function spaces. Diss. Math., 2017, 525: 1-102

[53] Song L, Yan L. Riesz transforms associated to Schrödinger operators on weighted Hardy spaces. J. Funct. Anal., 2010, 259: 1466-1490

[54] Stein EM , Weiss G. On the theory of harmonic functions of several variables. I. The theory of Hp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^p$$\end{document}-spaces. Acta Math., 1960, 103: 25-62

[55] Sun J, Yang D, Yuan W. Molecular characterization of weak Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type with its applications to Littlewood–Paley function characterizations. Forum Math., 2022, 34: 1539-1589

[56] Sun J , Yang D, Yuan W. Weak Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type: decompositions, real interpolation, and Calderón–Zygmund operators. J. Geom. Anal., 2022, 32: 191

[57] Wang F, Yang D, Yang S. Applications of Hardy spaces associated with ball quasi-Banach function spaces. Results Math., 2020, 75: 26

[58] Wang F, Yang D, Yuan W. Riesz transform characterization of Hardy spaces associated with ball quasi-Banach function spaces. J. Fourier Anal. Appl., 2023, 29: 56

[59] Wang S, Yang D, Yuan W, Zhang Y. Weak Hardy-type spaces associated with ball quasi-Banach function spaces II: Littlewood–Paley characterizations and real interpolation. J. Geom. Anal., 2021, 31: 631-696

[60] Yan L. Classes of Hardy spaces associated with operators, duality theorem and applications. Trans. Am. Math. Soc., 2008, 360: 4383-4408

[61] Yan X, He Z, Yang D, Yuan W. 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

[62] Yan X, He Z, Yang D, Yuan W. 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

[63] Yang D, Yang S. Musielak–Orlicz–Hardy spaces associated with operators and their applications. J. Geom. Anal., 2014, 24: 495-570

[64] Yang D, Zhang J. Variable Hardy spaces associated with operators satisfying Davies–Gaffney estimates on metric measure spaces of homogeneous type. Ann. Acad. Sci. Fenn. Math., 2018, 43: 47-87

[65] Yang D, Zhang J, Zhuo C. Variable Hardy spaces associated with operators satisfying Davies–Gaffney estimates. Proc. Edinb. Math. Soc. (2), 2018, 61: 759-810

[66] Yang D, Zhuo C. Molecular characterizations and dualities of variable exponent Hardy spaces associated with operators. Ann. Acad. Sci. Fenn. Math., 2016, 41: 357-398

[67] Zhang Y, Yang D, Yuan W, Wang S. Weak Hardy-type spaces associated with ball quasi-Banach function spaces I: decompositions with applications to boundedness of Calderón–Zygmund operators. Sci. China Math., 2021, 64: 2007-2064

[68] Zhu, C., Yang, D., Yuan, W.: Generalized Brezis–Seeger–Van Schaftingen–Yung formulae and their applications in ball Banach Sobolev spaces. Calc. Var. Partial Differential Equations 62, 234 (2023)

[69] Zhu, C., Yang, D., Yuan, W.: Extension theorem and Bourgain–Brezis–Mironescu type characterization of ball Banach Sobolev spaces on domains, Submitted for publication

[70] Zhu, C., Yang, D., Yuan, W.: Brezis–Seeger–Van Schaftingen–Yung-type characterization of homogeneous ball Banach Sobolev spaces and its applications. Commun. Contemp. Math. (2023). https://doi.org/10.1142/S0219199723500414

[71] Zhuo C, Sawano Y, Yang D. Hardy spaces with variable exponents on RD-spaces and applications. Diss. Math., 2016, 520: 74

Funding

National Key Research and Development Program of China(2020YFA0712900)

National Natural Science Foundation of China(11971058, 12071197, 12122102 and 12071431)

Fundamental Research Funds for the Central Universities(lzujbky-2021-ey18)

Innovative Groups of Basic Research in Gansu Province(22JR5RA391)

Key Project of Gansu Provincial National Science Foundation(23JRRA1022)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature