## Abstract

Let (X, d, μ) be the space of homogeneous type and Ω be a measurable subset of X which may not satisfy the doubling condition. Let L denote a nonnegative self-adjoint operator on L^{2}(Ω) which has a Gaussian upper bound on its heat kernel. The aim of this paper is to introduce a Hardy space HL1(Ω) associated to L on Ω which provides an appropriate setting to obtain HL1(Ω)→L1(Ω) boundedness for certain singular integrals with rough kernels. This then implies L^{p} boundedness for the rough singular integrals, 1 < p≤ 2 , from interpolation between the spaces L^{2}(Ω) and HL1(Ω). As applications, we show the boundedness for the holomorphic functional calculus and spectral multipliers of the operator L from HL1(Ω) to L^{1}(Ω) and on L^{p}(Ω) for 1 < p< ∞. We also study the case of the domains with finite measure and the case of the Gaussian upper bound on the semigroup replaced by the weaker assumption of the Davies–Gaffney estimate.

Original language | English |
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Journal | Collectanea Mathematica |

DOIs | |

State | Accepted/In press - 2022 |

## Keywords

- Hardy space on general domain
- Holomorphic functional calculus
- Interpolation
- Spectral multiplier