### Abstract

Let L_{1} and L_{2} be nonnegative self-adjoint operators acting on L^{2} (X_{1} ) and L^{2} (X_{2} ), respectively, where X_{1} and X_{2} are spaces of homogeneous type. Assume that L_{1} and L_{2} have Gaussian heat kernel bounds. This paper aims to study some equivalent characterizations of the weighted product Hardy spaces H^{p}_{w,L}_{1}_{,L2} (X_{1} × X_{2}) associated to L_{1} and L_{2} , for p ∈ (0, ∞) and the weight w belongs to the product Muckenhoupt class A_{∞}(X_{1} × X_{2}). Our main result is that the spaces H^{p}_{w,L}_{1},_{L2} (X_{1} × X_{2}) introduced via area functions can be equivalently characterized by the Littlewood–Paley g-functions and g^{∗}_{λ}_{1},_{λ2} -functions, as well as the Peetre type maximal functions, without any further assumption beyond the Gaussian upper bounds on the heat kernels of L_{1} and L_{2} . Our results are new even in the unweighted product setting.

Language | English |
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Pages | 91-115 |

Number of pages | 25 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 71 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

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**Equivalence of Littlewood–Paley square function and area function characterizations of weighted product Hardy spaces associated to operators.** / Duong, Xuan Thinh; Hu, Guorong; Li, Ji.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Equivalence of Littlewood–Paley square function and area function characterizations of weighted product Hardy spaces associated to operators

AU - Duong, Xuan Thinh

AU - Hu, Guorong

AU - Li, Ji

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Let L1 and L2 be nonnegative self-adjoint operators acting on L2 (X1 ) and L2 (X2 ), respectively, where X1 and X2 are spaces of homogeneous type. Assume that L1 and L2 have Gaussian heat kernel bounds. This paper aims to study some equivalent characterizations of the weighted product Hardy spaces Hpw,L1,L2 (X1 × X2) associated to L1 and L2 , for p ∈ (0, ∞) and the weight w belongs to the product Muckenhoupt class A∞(X1 × X2). Our main result is that the spaces Hpw,L1,L2 (X1 × X2) introduced via area functions can be equivalently characterized by the Littlewood–Paley g-functions and g∗λ1,λ2 -functions, as well as the Peetre type maximal functions, without any further assumption beyond the Gaussian upper bounds on the heat kernels of L1 and L2 . Our results are new even in the unweighted product setting.

AB - Let L1 and L2 be nonnegative self-adjoint operators acting on L2 (X1 ) and L2 (X2 ), respectively, where X1 and X2 are spaces of homogeneous type. Assume that L1 and L2 have Gaussian heat kernel bounds. This paper aims to study some equivalent characterizations of the weighted product Hardy spaces Hpw,L1,L2 (X1 × X2) associated to L1 and L2 , for p ∈ (0, ∞) and the weight w belongs to the product Muckenhoupt class A∞(X1 × X2). Our main result is that the spaces Hpw,L1,L2 (X1 × X2) introduced via area functions can be equivalently characterized by the Littlewood–Paley g-functions and g∗λ1,λ2 -functions, as well as the Peetre type maximal functions, without any further assumption beyond the Gaussian upper bounds on the heat kernels of L1 and L2 . Our results are new even in the unweighted product setting.

UR - http://www.scopus.com/inward/record.url?scp=85060729232&partnerID=8YFLogxK

U2 - 10.2969/jmsj/78287828

DO - 10.2969/jmsj/78287828

M3 - Article

VL - 71

SP - 91

EP - 115

JO - Journal of the Mathematical Society of Japan

T2 - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 1

ER -