## Abstract

Let χ be a space of homogeneous type. The aims of this paper are as follows. i) Assuming that T is a bounded linear operator on L_{2}(χ), we give a sufficient condition on the kernel of T so that T is of weak type (1,1), hence bounded on L_{p}(χ) for 1 < p ≤ 2; our condition is weaker than the usual Hörmander integral condition. ii) Assuming that T is a bounded linear operator on L_{2}(Ω) where Ω is a measurable subset of χ, we give a sufficient condition on the kernel of T so that T is of weak type (1,1), hence bounded on L_{p}(Ω) for 1 < p ≤ 2. iii) We establish sufficient conditions for the maximal truncated operator T_{*}, which is defined by T_{*}u(Greek cursive chi) = sup_{ε>0}|T_{ε}u(Greek curve chi)|, to be L_{p} bounded, 1 < p < ∞. Applications include weak (1, 1) estimates of certain Riesz transforms, and L_{p} boundedness of holomorphic functional calculi of linear elliptic operators on irregular domains.

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

Pages (from-to) | 233-265 |

Number of pages | 33 |

Journal | Revista Matematica Iberoamericana |

Volume | 15 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1999 |