## Abstract

Let (X, d, μ) be a metric measure space endowed with a metric d and a nonnegative Borel doubling measure μ. Let L be a non-negative self-adjoint operator of order m on X. Assume that L generates a holomorphic semigroup e ^{-tL} whose kernels pt(x, y) satisfy Gaussian upper bounds but without any assumptions on the regularity of space variables x and y. Also assume that L satisfies a Plancherel type estimate. Under these conditions, we show the L^{p} bounds for Stein's square functions arising from Bochner-Riesz means associated to the operator L. We then use the L^{p} estimates on Stein's square functions to obtain a Hörmander-type criterion for spectral multipliers of L. These results are applicable for large classes of operators including sub-Laplacians acting on Lie groups of polynomial growth and Schrödinger operators with rough potentials.

Original language | English |
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Pages (from-to) | 389-409 |

Number of pages | 21 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 65 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2013 |