Multi-Dimensional Hardy Spaces

  • Ferenc Weisz
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


In this chapter, two types of the multi-dimensional classical Hardy spaces, namely the \(H_{p}^{\square }(\mathbb{R}^{d})\) and \(H_{p}(\mathbb{R}^{d})\) spaces, are introduced. All the results of Chap.  1, amongst others, inequalities, atomic decompositions, interpolation theorems, boundedness results are proved for these spaces. Basically, the results for \(H_{p}^{\square }(\mathbb{R}^{d})\) are very similar to those for the one-dimensional \(H_{p}(\mathbb{R})\) spaces studied in Chap.  1, so we omit the corresponding proofs. However, the proofs for \(H_{p}(\mathbb{R}^{d})\) are different from the one-dimensional version requiring new ideas. We also study some generalizations of the Hardy-Littlewood maximal function for multi-dimensional functions.


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    A. Procházka, N.G. Kingsbury, P.J. Payner, and J. Uhlir: Signal Analysis and Prediction (ISBN 978-0-8176-4042-2)Google Scholar
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  35. 35.
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    M. An, A.K. Brodzik, and R. Tolimieri: Ideal Sequence Design in Time-Frequency Space (ISBN 978-0-8176-4737-7)Google Scholar
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  40. 40.
    B. Luong: Fourier Analysis on Finite Abelian Groups (ISBN 978-0-8176-4915-9)Google Scholar
  41. 41.
    G.S. Chirikjian: Stochastic Models, Information Theory, and Lie Groups, Volume 1 (ISBN 978-0-8176-4802-2)Google Scholar
  42. 42.
    C. Cabrelli and J.L. Torrea: Recent Developments in Real and Harmonic Analysis (ISBN 978-0-8176-4531-1)Google Scholar
  43. 43.
    M.V. Wickerhauser: Mathematics for Multimedia (ISBN 978-0-8176-4879-4)Google Scholar
  44. 44.
    B. Forster, P. Massopust, O. Christensen, K. Gröchenig, D. Labate, P. Vandergheynst, G. Weiss, and Y. Wiaux: Four Short Courses on Harmonic Analysis (ISBN 978-0-8176-4890-9)Google Scholar
  45. 45.
    O. Christensen: Functions, Spaces, and Expansions (ISBN 978-0-8176-4979-1)Google Scholar
  46. 46.
    J. Barral and S. Seuret: Recent Developments in Fractals and Related Fields (ISBN 978-0-8176-4887-9)Google Scholar
  47. 47.
    O. Calin, D.-C. Chang, and K. Furutani, and C. Iwasaki: Heat Kernels for Elliptic and Sub-elliptic Operators (ISBN 978-0-8176-4994-4)Google Scholar
  48. 48.
    C. Heil: A Basis Theory Primer (ISBN 978-0-8176-4686-8)Google Scholar
  49. 49.
    J.R. Klauder: A Modern Approach to Functional Integration (ISBN 978-0-8176-4790-2)Google Scholar
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    J. Cohen and A.I. Zayed: Wavelets and Multiscale Analysis (ISBN 978-0-8176-8094-7)Google Scholar
  51. 51.
    D. Joyner and J.-L. Kim: Selected Unsolved Problems in Coding Theory (ISBN 978-0-8176-8255-2)Google Scholar
  52. 52.
    G.S. Chirikjian: Stochastic Models, Information Theory, and Lie Groups, Volume 2 (ISBN 978-0-8176-4943-2)Google Scholar
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    J.A. Hogan and J.D. Lakey: Duration and Bandwidth Limiting (ISBN 978-0-8176-8306-1)Google Scholar
  54. 54.
    G. Kutyniok and D. Labate: Shearlets (ISBN 978-0-8176-8315-3)Google Scholar
  55. 55.
    P.G. Casazza and P. Kutyniok: Finite Frames (ISBN 978-0-8176-8372-6)Google Scholar
  56. 56.
    V. Michel: Lectures on Constructive Approximation (ISBN 978-0-8176-8402-0)Google Scholar
  57. 57.
    D. Mitrea, I. Mitrea, M. Mitrea, and S. Monniaux: Groupoid Metrization Theory (ISBN 978-0-8176-8396-2)Google Scholar
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    T.D. Andrews, R. Balan, J.J. Benedetto, W. Czaja, and K.A. Okoudjou: Excursions in Harmonic Analysis, Volume 1 (ISBN 978-0-8176-8375-7)Google Scholar
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    D.V. Cruz-Uribe and A. Fiorenza: Variable Lebesgue Spaces (ISBN 978-3-0348-0547-6)Google Scholar
  61. 61.
    W. Freeden and M. Gutting: Special Functions of Mathematical (Geo-)Physics (ISBN 978-3-0348-0562-9)Google Scholar
  62. 62.
    A. Saichev and W.A. Woyczynski: Distributions in the Physical and Engineering Sciences, Volume 2: Linear and Nonlinear Dynamics of Continuous Media (ISBN 978-0-8176-3942-6)Google Scholar
  63. 63.
    S. Foucart and H. Rauhut: A Mathematical Introduction to Compressive Sensing (ISBN 978-0-8176-4947-0)Google Scholar
  64. 64.
    G. Herman and J. Frank: Computational Methods for Three-Dimensional Microscopy Reconstruction (ISBN 978-1-4614-9520-8)Google Scholar
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    A. Paprotny and M. Thess: Realtime Data Mining: Self-Learning Techniques for Recommendation Engines (ISBN 978-3-319-01320-6)Google Scholar
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    A. Zayed and G. Schmeisser: New Perspectives on Approximation and Sampling Theory: Festschrift in Honor of Paul Butzer’s 85 th Birthday (978-3-319-08800-6)Google Scholar
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    H. Boche, R. Calderbank, G. Kutyniok, J. Vybiral: Compressed Sensing and its Applications (ISBN 978-3-319-16041-2)Google Scholar
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    S. Dahlke, F. De Mari, P. Grohs, and D. Labate: Harmonic and Applied Analysis: From Groups to Signals (ISBN 978-3-319-18862-1)Google Scholar
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    G. Pfander: Sampling Theory, a Renaissance (ISBN 978-3-319-19748-7)Google Scholar
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    R. Balan, M. Begue, J. Benedetto, W. Czaja, and K.A Okoudjou: Excursions in Harmonic Analysis, Volume 4 (ISBN 978-3-319-20187-0)Google Scholar
  72. 72.
    O. Christensen: An Introduction to Frames andRieszBases, Second Edition (ISBN 978-3-319-25611-5)Google Scholar
  73. 73.
    E. Prestini: The Evolution of Applied Harmonic Analysis:Models of the Real World, Second Edition (ISBN 978-1-4899-7987-2)Google Scholar
  74. 74.
    J.H. Davis: Methods of Applied Mathematics with a Software Overview, Second Edition (ISBN 978-3-319-43369-1)Google Scholar
  75. 75.
    M. Gilman, E. M. Smith, S. M. Tsynkov: Transionospheric Synthetic Aperture Imaging (ISBN 978-3-319-52125-1)Google Scholar
  76. 76.
    S. Chanillo, B. Franchi, G. Lu, C. Perez, E.T. Sawyer: Harmonic Analysis, Partial Differential Equations and Applications (ISBN 978-3-319-52741-3)Google Scholar
  77. 77.
    R. Balan, J. Benedetto, W. Czaja, M. Dellatorre, and K.A Okoudjou: Excursions in Harmonic Analysis, Volume 5 (ISBN 978-3-319-54710-7)Google Scholar
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    I. Pesenson, Q.T. Le Gia, A. Mayeli, H. Mhaskar, D.X. Zhou: Frames and Other Bases in Abstract and Function Spaces: Novel Methods in Harmonic Analysis, Volume 1 (ISBN 978-3-319-55549-2)Google Scholar
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    I. Pesenson, Q.T. Le Gia, A. Mayeli, H. Mhaskar, D.X. Zhou: Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science: Novel Methods in Harmonic Analysis, Volume 2 (ISBN 978-3-319-55555-3)Google Scholar
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    F. Weisz: Convergence and Summability of Fourier Transforms and Hardy Spaces (ISBN 978-3-319-56813-3)Google Scholar
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    For an up-to-date list of ANHA titles, please visit

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Ferenc Weisz
    • 1
  1. 1.Department of Numerical AnalysisEötvös Loránd UniversityBudapestHungary

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