caa a22 4500 477077005 CHVBK 20180405111441.0 cr unu---uuuuu 170330e19960101xx s 000 0 eng 10.1007/BF00049421 doi (NATIONALLICENCE)springer-10.1007/BF00049421 Subelliptic operators on Lie groups: Variable coefficients [Elektronische Daten] [Ola Bratteli, Derek Robinson] 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. Kluwer Academic Publishers, 1996 Lie groups nationallicence elliptic operators nationallicence subelliptic operators nationallicence parametrix method nationallicence holomorphy nationallicence Bratteli Ola Department of Mathematics, University of Oslo, Blindern, PB 1053, N-0316, Oslo 3, Norway aut Robinson Derek Centre for Mathematics and its Applications, Australian National University, 0200, Canberra, ACT, Australia aut Acta Applicandae Mathematica Kluwer Academic Publishers; gopher://gopher.wkap.nl 42/1(1996-01-01), 1-104 0167-8019 42:1<1 1996 42 10440 https://doi.org/10.1007/BF00049421 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF00049421 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Bratteli Ola Department of Mathematics, University of Oslo, Blindern, PB 1053, N-0316, Oslo 3, Norway aut NATIONALLICENCE 700 1- Robinson Derek Centre for Mathematics and its Applications, Australian National University, 0200, Canberra, ACT, Australia aut NATIONALLICENCE 773 0- Acta Applicandae Mathematica Kluwer Academic Publishers; gopher://gopher.wkap.nl/ 42/1(1996-01-01), 1-104 0167-8019 42:1<1 1996 42 10440 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer