Invariant Differential Operators Research Articles

Convolution equation and operators on the euclidean motion group

Let $G = \mathbb{R}^2\rtimes SO(2)$ be the Euclidean motion group, let g be the Lie algebra of G and let U(g) be the universal enveloping algebra of g. Then U(g) is an infinite dimensional, linear associative and non-commutative algebra consisting of invariant differential operators on G. The Dirac measure on G is represented by $\delta_G$, while the convolution product of functions or measures on G is represented by $\ast$. Among other notable results, it is demonstrated that for each u in U(g), there is a distribution E on G such that the convolution equation $u\ast E = \delta_G$ is solved by method of convolution. Further more, it is established that the (convolution) operator $A^\prime : C^\infty_c(G)\rightarrow C^\infty(G),$, which is defined as $A^\prime f = f\ast T^n\delta(t)$ extends to a bounded linear operator on $L^2(G)$, for $f\in C^\infty_c(G)$, the space of infinitely differentiable functions on G with compact support. Furthermore, we demonstrate that the left convolution operator LT denoted as $L_Tf = T\ast f$ commutes with left translation, for $T\in D^\prime(G)$.

Construction of first-order invariant differential operators

The paper considers the problem of constructing systems of vector fields that are invariant under the action of the local Lie group of transformations. It is shown that there exists a special class of Lie groups for which this problem can be solved elementarily.

Construction of First-Order Invariant Differential Operators

Construction of First-Order Invariant Differential Operators

Hypoelliptic functional inequalities

In this paper we derive a variety of functional inequalities for general homogeneous invariant hypoelliptic differential operators on nilpotent Lie groups. The obtained inequalities include Hardy, Sobolev, Rellich, Hardy–Littllewood–Sobolev, Gagliardo–Nirenberg, Caffarelli–Kohn–Nirenberg and Heisenberg–Pauli–Weyl type uncertainty inequalities. Some of these estimates have been known in the case of the sub-Laplacians, however, for more general hypoelliptic operators almost all of them appear to be new as no approaches for obtaining such estimates have been available. The approach developed in this paper relies on establishing integral versions of Hardy inequalities on homogeneous Lie groups, for which we also find necessary and sufficient conditions for the weights for such inequalities to be true. Consequently, we link such integral Hardy inequalities to different hypoelliptic inequalities by using the Riesz and Bessel kernels associated to the described hypoelliptic operators.

Breaking symmetries for equivariant coarse homology theories

We describe a symmetry breaking construction in coarse geometry which allows to obtain information about equivariant coarse homology classes by restriction to smaller groups and spaces. In the case of equivariant coarse K-homology theory we give an analytic interpretation of this construction. As a consequence we obtain applications to the spectral theory of invariant differential operators.

Conformally invariant differential operators on Heisenberg groups and minimal representations

Conformally invariant differential operators on Heisenberg groups and minimal representations

Invariant differential operators for noncompact Lie groups: The Sp(n,1) case

In this paper, we continue the project of systematic construction of invariant differential operators (IDOs) on the example of the noncompact algebras [Formula: see text]. Our choice of these algebras is motivated by the fact that they belong to a narrow class of algebras, which are of split rank one, of which class the other cases were studied, some long time ago. We concentrate on the case [Formula: see text]. We give the main multiplets and the main reduced multiplets of indecomposable elementary representations (ER) for, including the necessary data for all relevant invariant differential operators. We also present explicit expressions for the singular vectors and the intertwining differential operators.

Canonical Construction of Invariant Differential Operators: A Review

In the present paper, we review the progress of the project of the classification and construction of invariant differential operators for non-compact, semisimple Lie groups. Our starting point is the class of algebras which we called earlier ‘conformal Lie algebras’ (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this purpose, we introduced recently the new notion of a parabolic relation between two non-compact, semi-simple Lie algebras G and G′ that have the same complexification and possess maximal parabolic subalgebras with the same complexification.

Einstein Manifold of SO(p,q) with a New Parabolic Subgroup and Invariant Differential Operators

In the present paper we continue the project of systematic explicit construction of invariant differential operators. On the example of the non-compact group SO(p, 9 − p) (for p = 5, 6) we give the classification multiplets of indecomposable elementary representations induced from a new (relative to earlier considerations) choice of parabolic subgroup MAN so that the factor-group G/M is an Einstein manifold. This classification includes the data for the relevant invariant differential operators.

Analysis of the critical CR GJMS operator

Abstract: The critical CR GJMS operator on a strictly pseudoconvex CR manifold is a non-hypoelliptic CR invariant differential operator. We prove that, under the embeddability assumption, it is essentially self-adjoint and has closed range. Moreover, its spectrum is discrete, and the eigenspace corresponding to each non-zero eigenvalue is a finite-dimensional subspace of the space of smooth functions. As an application, we obtain a necessary and sufficient condition for the existence of a contact form with zero CR $Q$-curvature.

Einstein Manifold of SO(p, 9 – p), Its Corresponding Parabolic Subgroup, and Invariant Differential Operators

In the present paper we continue the project of systematic explicit construction of invariant differential operators. On the example of the non-compact group SO(p,9 - p) (for p = 5,6) we give the classification multiplets of indecomposable elementary representations induced from suitable choice of parabolic subgroup MAN so that the factor-group {G mathord{left/ {vphantom {G M}} right. kern-0em} M} is an Einstein manifold. This classification includes the data for the relevant invariant differential operators.

Generalized Willmore energies, Q-curvatures, extrinsic Paneitz operators, and extrinsic Laplacian powers

Over forty years ago, Paneitz, and independently Fradkin and Tseytlin, discovered a fourth-order conformally invariant differential operator, intrinsically defined on a conformal manifold, mapping scalars to scalars. This operator is a special case of the so-termed extrinsic Paneitz operator defined in the case when the conformal manifold is itself a conformally embedded hypersurface. In particular, this encodes the obstruction to smoothly solving the five-dimensional scalar Laplace equation, and suitable higher dimensional analogs, on conformally compact structures with constant scalar curvature. Moreover, the extrinsic Paneitz operator can act on tensors of general type by dint of being defined on tractor bundles. Motivated by a host of applications, we explicitly compute the extrinsic Paneitz operator. We apply this formula to obtain: an extrinsically-coupled [Formula: see text]-curvature for embedded four-manifolds, the anomaly in renormalized volumes for conformally compact five-manifolds with negative constant scalar curvature, Willmore energies for embedded four-manifolds, the local obstruction to smoothly solving the five-dimensional singular Yamabe problem, and new extrinsically-coupled fourth- and sixth-order operators for embedded surfaces and four-manifolds, respectively.

Einstein Manifold of Pseudo-Orthogonal Group and Invariant Differential Operators

Einstein Manifold of Pseudo-Orthogonal Group and Invariant Differential Operators

Curved Versions of the Ovsienko–Redou Operators

Abstract We construct a family of conformally covariant bidifferential operators on pseudo-Riemannian manifolds. Our construction is analogous to the construction of Graham–Jenne–Mason–Sparling of conformally covariant differential operators via tangential powers of the Laplacian in the Fefferman–Graham ambient space. In fact, we completely classify the tangential bidifferential operators on the ambient space, which are expressed purely in terms of the ambient Laplacian. This gives a curved analogue of the classification, due to Ovsienko–Redou and Clerc, of conformally invariant bidifferential operators on the sphere. As an application, we construct a large class of formally self-adjoint conformally invariant differential operators.

The Fourth-Linear aff(1)-Invariant Differential Operators and the First Cohomology of the Lie Algebra of Vector Fields on RP1

In this paper, we denote the Lie algebra of smooth vector fields on RP1 by V(RP1). This paper focuses on two parts. In the beginning, we determine the cohomology space of aff(1) with coefficients in Dτ,λ,μ;ν. Afterward, we classify aff(1)-invariant fourth-linear differential operators from V(RP1) to Dτ,λ,μ;ν vanishing on aff(1). This result enables us to compute the aff(1)-relative cohomology of V(RP1) with coefficients in Dτ,λ,μ;ν.

Invariant bilinear differential operators

I classified bilinear differential operators acting in the spaces of tensor fields on any real or complex manifold and invariant with respect to the diffeomorphisms in 1980. Here I give the details of the proof.

On [formula omitted]-relative cohomology on [formula omitted

Over the (1,2)-dimensional real superspace R1|2. We compute the second aff(2|1)-relative cohomology space of K(2) with coefficients in the module of λ-densities Fλ2 on R1|2, where K(2) is the Lie superalgebra of contact vector fields on R1|2 and aff(2|1) is the affine Lie superalgebra. We give explicitly 2-cocycles spanning these cohomology spaces. This result is based specially on the classification of aff(2|1)-invariant bilinear differential operators acting on the superspaces of weighted densities given in [2].

Delorme’s intertwining conditions for sections of homogeneous vector bundles on two- and three-dimensional hyperbolic spaces

The description of the Paley–Wiener space for compactly supported smooth functions \(C^\infty _c(G)\) on a semi-simple Lie group G involves certain intertwining conditions that are difficult to handle. In the present paper, we make them completely explicit for \(G=\textbf{SL}(2,\mathbb {R})^d\) (\(d\in \mathbb {N}\)) and \(G=\textbf{SL}(2,\mathbb {C})\). Our results are based on a defining criterion for the Paley–Wiener space, valid for general groups of real rank one, that we derive from Delorme’s proof of the Paley–Wiener theorem. In a forthcoming paper, we will show how these results can be used to study solvability of invariant differential operators between sections of homogeneous vector bundles over the corresponding symmetric spaces.

Classification of 𝔞𝔣𝔣(3|1)-invariant binary differential operators on weighted densities and cohomology

In this paper, we consider the [Formula: see text]-module structure on the spaces of differential bilinear operators acting on the superspaces of weighted densities. We classify [Formula: see text]-invariant binary differential operators acting on the spaces of weighted densities. This result allows us to compute the first [Formula: see text]-relative differential cohomology of [Formula: see text] with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. Finally, we deduce the classification of the [Formula: see text]-relative extension of [Formula: see text] modules [Formula: see text]

Polynomial null solutions to bosonic Laplacians, bosonic Bergman and Hardy spaces

AbstractA bosonic Laplacian, which is a generalization of Laplacian, is constructed as a second-order conformally invariant differential operator acting on functions taking values in irreducible representations of the special orthogonal group, hence of the spin group. In this paper, we firstly introduce some properties for homogeneous polynomial null solutions to bosonic Laplacians, which give us some important results, such as an orthogonal decomposition of the space of polynomials in terms of homogeneous polynomial null solutions to bosonic Laplacians, etc. This work helps us to introduce Bergman spaces related to bosonic Laplacians, named as bosonic Bergman spaces, in higher spin spaces. Reproducing kernels for bosonic Bergman spaces in the unit ball and a description of bosonic Bergman projection are given as well. At the end, we investigate bosonic Hardy spaces, which are considered as generalizations of harmonic Hardy spaces. Analogs of some well-known results for harmonic Hardy spaces are provided here. For instance, connections to certain complex Borel measure spaces, growth estimates for functions in the bosonic Hardy spaces, etc.