A functional calculus of pseudodifferential operators on unimodular Lie groups

An algebra of pseudodifferential operators on a unimodular Lie group is constructed. It is defined by uniform estimates of the local symbols and by conditions on the decrease of the kernel outside the diagonal, formulated in terms of the weight function, having an intermediate growth between the volume function and a standard exponential. The decrease of the Green function is described, complex powers of elliptic operators of the considered classes are constructed, a meromorphic continuation of the kernels of complex powers is described, from where, in particular, the asymptotic behavior of the spectral function of operators of the considered class is obtained.Translated from Trudy Seminara im. I. G. Petrovskogo, No. 12, pp. 164–200, 1987.     

On a class of pseudodifferential operators on the product of compact Lie groups

In this paper, a bisingular pseudodifferential calculus, along the lines of the one introduced by L. Rodino in his paper of 1975, is developed in the global setting of a product of compact Lie groups. The approach follows that introduced by M. Ruzhansky and V. Turunen in their book of 2010 (see also V. Fischer's paper of 2015), in that it exploits the harmonic analysis of the groups involved.     

Metrics on 4-dimensional unimodular Lie groups

We classify left invariant metrics on the 4-dimensional, simply connected, unimodular Lie groups up to automorphism. When the corresponding Lie algebra is of type (R), this is equivalent to classifying the left invariant metrics up to isometry, but in general the classification up to automorphism is finer than that up to isometry. In the abelian case, all left invariant metrics are isometric. In the nilpotent case, the space of metrics can have dimension 1 or 3. In the solvable case, the dimension can be 2, 4, or 5. There are two non-solvable 4-dimensional unimodular groups, and the space of metrics has dimension 6 in both of these cases.     

Elliptic operators on Lie groups

We review the theory of strongly elliptic operators on Lie groups and describe some new simplifications. Let U be a continuous representation of a Lie group G on a Banach space and a ₁,...,a _d a basis of the Lie algebra g of G. Let A _i=dU(a _i) denote the infinitesimal generator of the continuous one-parameter group t U(exp(-ta _i)) and set % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0df9GqVa0-% aq0dXdarVe0-yr0RYxir-dbba9q8aq0-qq-He9q8qqQ8fq0-vr0-vr% Y-bdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaCaaale% qajeaObaGaeyySdegaaOGaeyypa0JaamyqamaaBaaajeaWbaGaaeyA% aaWcbeaajaaOdaWgaaqcbaAaamaaBaaajiaObaGaaiiBaaqabaaaje% aObeaakiaacElacaGG3cGaai4TaiaadgeadaWgaaqcbaCaaiaabMga% aSqabaGcdaWgaaWcbaWaaSbaaKGaahaacaGGUbaameqaaaWcbeaaaa% a!4897!\[A^\alpha = A_{\rm{i}} _{_l } \cdot\cdot\cdotA_{\rm{i}} _{_n } \], where =(i ₁,...,i _n) with _j and set ||=n. We analyze properties of mth order differential operators % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0df9GqFj0-% aq0dXdarVe0-yr0RYxir-dbba9q8aq0-qq-He9q8qqQ8fq0-vr0-vr% Y-bdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiabg2da9i% aabccadaaeqaqaaiaadogadaWgaaqcbaCaaiabgg7aHbWcbeaaaKqa% GgaacqGHXoqycaqG7aGaaeiiaiaabYhacqGHXoqycaqG8bGaeyizIm% QaaeyBaaWcbeqdcqGHris5aOGaamyqamaaCaaaleqajeaObaGaeyyS% degaaaaa!4A6C!\[H = {\rm{ }}\sum\nolimits_{\alpha {\rm{; |}}\alpha {\rm{|}} \le {\rm{m}}} {c_\alpha } A^\alpha \] with coefficients c . If H is strongly elliptic, i.e., % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0df9GqFj0-% aq0dXdarVe0-yr0RYxir-dbba9q8aq0-qq-He9q8qqQ8fq0-vr0-vr% Y-bdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciOuaiaacwgacq% GH9aqpcaqGGaWaaabeaeaacaGGOaaajeaObaGaeyySdeMaae4oaiaa% bccacaqG8bGaeyySdeMaaeiFaiabg2da9iaab2gaaSqab0GaeyyeIu% oakiaabMgacqaH+oaEcaGGPaWaaWbaaSqabKqaGgaacqGHXoqyaaGc% cqGH+aGpcaaIWaaaaa!4C40!\[{\mathop{\rm Re}\nolimits} = {\rm{ }}\sum\nolimits_{\alpha {\rm{; |}}\alpha {\rm{|}} = {\rm{m}}} ( {\rm{i}}\xi )^\alpha > 0\] for all % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0df9GqVa0-% aq0dXdarVe0-yr0RYxir-dbba9q8aq0-qq-He9q8qqQ8fq0-vr0-vr% Y-bdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdGNaeyicI4% SaeSyhHe6aaWbaaSqabeaacaWGKbaaaOGaaiixaiaacUhacaaIWaGa% aiyFaaaa!3EAA!\[\xi \in ^d \backslash \{ 0\} \], then we give a simple proof of the theorem that the closure of H generates a continuous (and holomorphic) semigroup on and the action of the semigroup is determined by a smooth, representation independent, kernel which, together with all its derivatives, satisfies mth order Gaussian bounds.     

Unitary representations of unimodular Lie groups in Bergman spaces

For an arbitrary unimodular Lie group G, we construct strongly continuous unitary representations in the Bergman space of a strongly pseudoconvex neighborhood of G in the complexification of its underlying manifold. These representation spaces are infinite-dimensional and have compact kernels. In particular, the Bergman spaces of these natural manifolds are infinite-dimensional.     

Short-time asymptotics of heat kernels of hypoelliptic Laplacians on unimodular Lie groups

We consider the problem of computing heat kernel small-time asymptotics for hypoelliptic Laplacians associated to left-invariant sub-Riemannian structures on unimodular Lie groups of type I. We use the non-commutative Fourier transform of the Lie group together with perturbation theory for semigroups of operators in deriving these asymptotics. We illustrate our approach on the example of the Heisenberg group, and, as an application, we compute the short-time behaviour of the hypoelliptic heat kernel on the step 3 nilpotent Cartan and Engel groups, for which no closed-form expression for the hypoelliptic heat kernel is yet known.     

The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups

We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector using the Popp's volume form introduced by Montgomery. This definition generalizes the one of the Laplace-Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares.We then extend a method (first used by Hulanicki on the Heisenberg group) to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncommutative Fourier transform. We then study some relevant cases: SU(2), SO(3), SL(2) (with the metrics inherited by the Killing form), and the group SE(2) of rototranslations of the plane.     

Soliton metrics for two-loop renormalization group flow on 3D unimodular Lie groups

The two-loop renormalization group flow is studied via the induced bracket flow on 3D unimodular Lie groups. A number of steady solitons are found. Some of these steady solitons come from maximally symmetric metrics that are steady, shrinking, or expanding solitons under Ricci flow, while others are not obviously related to Ricci flow solitons.     

Non-local Dirichlet forms generated by pseudodifferential operators on compact abelian groups

Let G_i, 1 ≤ i ≤ n, be compact abelian groups and let Γ_i , 1 ≤ i ≤ n, be countable dual groups. We consider G = G₁ ⊕ G₂ ⊕ … ⊕ G_n and Γ = Γ₁ ⊕ Γ₂ ⊕ … ⊕ Γ_n . For 1 ≤ j ≤ n, let a_j be a negative definite function on Γ_j and a (γ) = . For φ ∈ S (G), the set of all generalized trigonometrical polynomials on G, we define , where (γ) = a_j (γ_j) (γ), 1 ≤ j ≤ n. Then is a Dirichlet form with the domain on L² (G). (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lower bounds for operators on graded Lie groups

In this note we present a symbolic pseudo-differential calculus on any graded (nilpotent) Lie group and, as an application, a version of the sharp Gårding inequality. As a corollary, we obtain lower bounds for positive Rockland operators with variable coefficients as well as their Schwartz-hypoellipticity.     

Subelliptic operators on Lie groups: Variable coefficients

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(a_i) 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 . 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 ; ||=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 d, 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 ||=2 are three times differentiable and those with ||=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.