## Abstract

Let (X,d,μ) be a metric measure space endowed with a distance d and a nonnegative Borel doubling measure μ. Let L be a non-negative self-adjoint operator on L^{2}(X). Assume that the semigroup e^{-tL} generated by L satisfies the Davies-Gaffney estimates. Let H_{L} ^{p}(X) be the Hardy space associated with L. We prove a Hörmander-type spectral multiplier theorem for L on H_{L} ^{p}(X) for 0 < p < ∞ the operator m(L) is bounded from H_{L}^{p}(X) to H_{L}^{p}(X) if the function m possesses s derivatives with suitable bounds and s > n(1/p - 1/2) where n is the "dimension" of X. By interpolation, m(L) is bounded on H _{L}^{p}(X) for all 0 < p < ∞ if m is infinitely differentiable with suitable bounds on its derivatives. We also obtain a spectral multiplier theorem on L^{p} spaces with appropriate weights in the reverse Hölder class.

Original language | English |
---|---|

Pages (from-to) | 295-319 |

Number of pages | 25 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 63 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2011 |