### Abstract

Let L=-δ+μ be the generalized Schrödinger operator on R^{n}, n≥3, where μ≢0 is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. Based on Shen's work for the fundamental solution of L in [23], we establish the following upper bound for semigroup kernels K_{t}(x,y), associated to e^{-tL},0≤K_{t}(x,y)≤Ch_{t}(x-y)e^{-εdμ(x,y,t)}, where h_{t}(x)=(4πt)^{-n/2}e-|x|^{2}/(4t), and d_{μ}(x, y, t) is some parabolic type distance function associated with μ. As a consequence,0≤K_{t}(x,y)≤Ch_{t}(x-y)exp (-c0(1+m(x,μ)max {|x-y|,√t})1/k_{0}+1),where m(x, μ) is some auxiliary function associated with μ. We then study a Hardy space H_{L}^{1} by means of a maximal function associated with the heat semigroup e^{-tL} generated by -L to obtain its characterizations via atomic decomposition and Riesz transforms. Also the dual space BMO_{L} of H_{L}^{1} is studied in this paper.

Original language | English |
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Pages (from-to) | 3709-3749 |

Number of pages | 41 |

Journal | Journal of Functional Analysis |

Volume | 270 |

Issue number | 10 |

DOIs | |

Publication status | Published - 15 May 2016 |

Externally published | Yes |

### Keywords

- hardy space
- heat kernel
- scale-invariant kato conditions
- schrödinger operators

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## Cite this

*Journal of Functional Analysis*,

*270*(10), 3709-3749. https://doi.org/10.1016/j.jfa.2015.12.016