Hardy spaces meet harmonic weights

Preisner, Marcin; Sikora, Adam; Yan, Lixin

doi:10.1090/tran/8695

Hardy spaces meet harmonic weights
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by Marcin Preisner, Adam Sikora and Lixin Yan PDF

Trans. Amer. Math. Soc. 375 (2022), 6417-6451 Request permission

Abstract:

We investigate the Hardy space $H^1_L$ associated with a self-adjoint operator $L$ defined in a general setting by Hofmann, Lu, Mitrea, Mitrea, and Yan [Mem. Amer. Math. Soc. 214 (2011), pp. vi+78]. We assume that there exists an $L$-harmonic non-negative function $h$ such that the semigroup $\exp (-tL)$, after applying the Doob transform related to $h$, satisfies the upper and lower Gaussian estimates. Under this assumption we describe an illuminating characterisation of the Hardy space $H^1_L$ in terms of a simple atomic decomposition associated with the $L$-harmonic function $h$. Our approach also yields a natural characterisation of the $BMO$-type space corresponding to the operator $L$ and dual to $H^1_L$ in the same circumstances.

The applications include surprisingly wide range of operators, such as: Laplace operators with Dirichlet boundary conditions on some domains in ${\mathbb {R}^n}$, Schrödinger operators with certain potentials, and Bessel operators.

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