## Four Lectures on Real Hp̳ SpacesThis book introduces the real variable theory of HP spaces briefly and concentrates on its applications to various aspects of analysis fields. It consists of four chapters. Chapter 1 introduces the basic theory of Fefferman-Stein on real HP spaces. Chapter 2 describes the atomic decomposition theory and the molecular decomposition theory of real HP spaces. In addition, the dual spaces of real HP spaces, the interpolation of operators in HP spaces, and the interpolation of HP spaces are also discussed in Chapter 2. The properties of several basic operators in HP spaces are discussed in Chapter 3 in detail. Among them, some basic results are contributed by Chinese mathematicians, such as the decomposition theory of weak HP spaces and its applications to the study on the sharpness of singular integrals, a new method to deal with the elliptic Riesz means in HP spaces, and the transference theorem of HP-multipliers etc. The last chapter is devoted to applications of real HP spaces to approximation theory. |

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### Contents

Preface Chapter 1 Real Variable Theory of HpRn Spaces 1 Definition of HpRn spaces | 1 |

2 Nontangential maximal functions | 3 |

3 Grand maximal functions | 13 |

Decomposition Structure Theory of HpRn Spaces 1 Atom | 20 |

2 Dual space of H 1Wl | 25 |

3 Atom decomposition | 34 |

4 Dual space of HpRn | 54 |

5 Interpolation of operators | 63 |

2 The Fourier multiplier | 98 |

3 The Riesz potential operators | 105 |

4 Singular integral operators | 108 |

5 The BochnerRiesz means | 121 |

6 Transference theorems of Hp multipliers | 144 |

Applications to Approximation Theory 1 K functional | 173 |

2 Hp multiplier and Jacksontype inequality | 175 |

3 Hp multiplier and Bernstein type inequality | 182 |

6 Interpolations of Hp spaces weak Hp spaces | 74 |

7 Molecule molecule decomposition | 83 |

8 Applications to the boundedness of operators | 90 |

Applications to Fourier Analysis 1 Fourier transform | 95 |

4 Approximation by BochnerRiesz means at critical index | 189 |

References | 214 |

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### Common terms and phrases

assume atomic decomposition BMO(R Bochner-Riesz means bounded mean oscillation bounded operator boundedness completes the proof conclusion of Theorem condition of atoms constant C independent Corollary critical index defined Definition 1.1 denote dominated convergence theorem dual spaces E. M. Stein E.M.Stein easy to verify equality exists a constant fact finishes the proof follows G Hp(Rn G.Weiss H.P.Liu Hardy inequality Hardy spaces homogeneous Hp multiplier Hp n L2 Hp(Tn immediately deduced inequality implies Lemma Lp(Rn Marcinkiewicz interpolation theorem maximal function maximal multiplier molecule multiplier on Hp(Rn nonnegative integer Note obtain Obviously order to prove Poisson kernel proof of Proposition proof of Theorem properties Proposition 3.3 prove the theorem Riesz means Riesz transform S(Rn satisfies singular integral operator Studia Math suffices to prove suffices to show supp Suppose Theorem 3.1 trigonometric polynomial vanishing moment condition weak Hp spaces weak type Z.X.Liu