## Abstract

Let G be a connected Lie group with Lie algebra g and a_{1}, . . . , a_{d′} an algebraic basis of g. Further let A_{i} denote the generators of left translations, acting on the L_{p}-spaces L_{p}(G ; dg) formed with left Haar measure dg, in the directions a_{i}. We consider second-order operators H = - ∑_{i,j=1}^{d′} A_{i}C_{ij}A_{j} + ∑_{i=1}^{d′} (c_{i}A_{i} + A_{i}c′_{i} + c_{0}I corresponding to a quadratic form with complex coefficients c_{ij}, c_{i}, c′_{i}, c_{0} ε L_{∞}. The principal coefficients C_{ij} are assumed to be Hölder continuous and the matrix C = (c_{ij}) is assumed to satisfy the (sub)ellipticity condition ℜC = 2^{-1}(C + C*) ≥ μI > 0 uniformly over G. We discuss the hierarchy relating smoothness properties of the coefficients of H with smoothness of the kernel. Moreover, we establish Gaussian type bounds for the kernel and its derivatives. Similar theorems are proved for operators H′ = - ∑_{i,j=1}^{d′} c_{ij}A_{i}A_{j} + ∑_{i=1}^{d′} c_{i}A_{i} + c_{0}I in nondivergence form for which the principal coefficients are at least once differentiable.

Original language | English |
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Pages (from-to) | 327-363 |

Number of pages | 37 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 8 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1999 |