ALBERT

Notes: Abstract Let G be a Lie group with Lie algebra g and a i,...,a d′ and algebraic basic of g. Futher, if A i=dL(ai) are the corresponding generators of left translations by G on one of the usual function spaces over G, let % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamXvP5wqonvsaeHbfv3ySLgzaGqbciab-Heaijaab2dadaaeqbqa% aiaadogadaWgaaWcbaqedmvETj2BSbacgmGae4xSdegabeaakiaadg% eadaahaaWcbeqaaiab+f7aHbaaaeaacqGFXoqycaGG6aGaaiiFaiab% +f7aHjaacYhatuuDJXwAK1uy0HMmaeXbfv3ySLgzG0uy0HgiuD3BaG% Wbbiab9rMiekaaikdaaeqaniabggHiLdaaaa!5EC1!\[H{\rm{ = }}\sum\limits_{\alpha :|\alpha | \le 2} {c_\alpha A^\alpha } \] be a second-order differential operator with real bounded coefficients c α. The operator is defined to be subelliptic if % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiGacMgacaGGUbGaaiOzamXvP5wqonvsaeHbfv3ySLgzaGqbaKaz% aasacqWF7bWEcqWFTaqlkmaaqafabaGaam4yamaaBaaaleaarmWu51% MyVXgaiyWacqGFXoqyaeqaaaqaaiab+f7aHjaacQdacaGG8bGae4xS% deMaaiiFaiabg2da9iaaikdaaeqaniabggHiLdGccqWFOaakiuGacq% qFNbWzcqWFPaqkcqaH+oaEdaahaaWcbeqaamaaBaaameaacqGFXoqy% aeqaaaaakiaacUdacqqFNbWzcqGHiiIZcqqFhbWrcqqFSaalcqqFGa% aicqaH+oaEcqGHiiIZrqqtubsr4rNCHbachaGaeWxhHe6aaWbaaSqa% beaacqqFKbazcqqFNaWjcqaFaC-jaaGccaGGSaGaaiiFaiabe67a4j% aacYhacqGH9aqpjqgaGeGae8xFa0NccqGH+aGpcaaIWaGaaiOlaaaa% !7884!\[\inf \{ - \sum\limits_{\alpha :|\alpha | = 2} {c_\alpha } (g)\xi ^{_\alpha } ;g \in G, \xi \in ^{d'} ,|\xi | = \} 〉 0.\] We prove that if the principal coefficients {c α; |α|=2} of the subelliptic operator are once left differentiable in the directions a 1,...,a d′ with bounded derivatives, then the operator has a family of semigroup generator extensions on the L p-spaces with respect to left Haar measure dg, or right Haar measure dĝ, and the corresponding semigroups S are given by a positive integral kernel, % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamXvP5wqonvsaeHbfv3ySLgzaGqbaiab-HcaOGqbciab+nfatnaa% BaaaleaacaWG0baabeaaruqqYLwySbacgiGccaqFgpGae8xkaKIae8% hkaGIae43zaCMae8xkaKIae8xpa0Zaa8qeaeaacaqGKbaaleaacqGF% hbWraeqaniabgUIiYdGcceWGObGbaKaacaWGlbWaaSbaaSqaaiaads% haaeqaaOGae8hkaGIae43zaCMae43oaSJae4hAaGMae8xkaKIaa0NX% diab-HcaOiab+HgaOjab-LcaPiab-5caUaaa!5DFA!\[(S_t \phi )(g) = \int_G {\rm{d}} \hat hK_t (g;h)\phi (h).\] The semigroups are holomorphic and the kernel satisfies Gaussian upper bounds. If in addition the coefficients with |α|=2 are three times differentiable and those with |α|=1 are once differentiable, then the kernel also satisfies Gaussian lower bounds. Some original features of this article are the use of the following: a priori inequalities on L ∞ in Section 3, fractional operator expansions for resolvent estimates in Section 4, a parametrix method based on reduction to constant coefficient operators on the Lie group rather than the usual Euclidean space in Section 5, approximation theory of semigroups in Section 11 and ‘time dependent’ perturbation theory to treat the lower order terms of H in Sections 11 and 12.