Subelliptic Estimates Research Articles

Subellipticity, compactness, $H^{\epsilon}$ estimates and regularity for $\bar{\partial}$ on weakly $q$-pseudoconvex/concave domains

For a weakly q -pseudoconvex (resp. q -pseudoconcave) domain \Omega in a Stein manifold X of dimension n , we give a sufficient condition for subelliptic estimates for the \bar{\partial} -Neumann problem. Moreover, we study the compactness of the \bar{\partial} -Neumann operator N on \Omega . Such compactness estimates immediately lead to smoothness of solutions, the closed range property, the L^{2} -setting and the Sobolev estimates of N on \Omega for any \bar{\partial} -closed (r, k) -form with k \geqslant q (resp. k \leqslant q ). Furthermore, we study the \bar{\partial} -problem with support conditions in \Omega for forms of type (r, k) , with values in a holomorphic vector bundle. Applications to the \bar{\partial}_{b} -problem for smooth forms on boundaries of \Omega are given.

Spectral stability of the Kohn Laplacian under perturbations of the boundary

In this paper, we study stability of the spectrum of the Kohn Laplacian □b on the boundary of a smoothly bounded domain in Cn as the boundary is perturbed in the C2-topology. We obtain estimates for spectral stability of the Kohn Laplacian on smooth compact hypersurfaces that satisfy uniform subelliptic estimate, in particular for strictly pseudoconvex hypersurfaces in Cn and pseudoconvex hypersurfaces of finite type in C2.

Polar decomposition of semigroups generated by non-selfadjoint quadratic differential operators and regularizing effects

We study semigroups generated by accretive non-selfadjoint quadratic differential operators. We give a description of the polar decomposition of the associated evolution operators as products of a selfadjoint operator and a unitary operator. The selfadjoint parts turn out to be also evolution operators generated by time-dependent real-valued quadratic forms that are studied in details. As a byproduct of this decomposition, we give a geometric description of the regularizing properties of semigroups generated by accretive non-selfadjoint quadratic operators. Finally, by using the interpolation theory, we take advantage of this smoothing effect to establish subelliptic estimates enjoyed by quadratic operators.

A remark on Kohn's theorem on sums of squares of complex vector fields

The plan of this paper is to give an alternate proof of Kohn's subelliptic estimate for systems of N smooth complex vector fields on an open set of Rn, and to improve it in extending the result to perturbations by a first-order term. A pseudodifferential generalization will also be given.

Subelliptic estimates from Gromov hyperbolicity

In this paper we prove: if the complete Kähler-Einstein metric on a bounded convex domain (with no boundary regularity assumptions) is Gromov hyperbolic, then the ∂¯-Neumann problem satisfies a subelliptic estimate. This is accomplished by constructing bounded plurisubharmonic function whose Hessian grows at a certain rate (which implies a subelliptic estimate by work of Catlin and Straube). We also provide a characterization of Gromov hyperbolicity in terms of orbit of the domain under the group of affine transformations. This characterization allows us to construct many examples. For instance, if the Hilbert metric on a bounded convex domain is Gromov hyperbolic, then the Kähler-Einstein metric is as well.

Description of the smoothing effects of semigroups generated by fractional Ornstein–Uhlenbeck operators and subelliptic estimates

We study semigroups generated by general fractional Ornstein–Uhlenbeck operators acting on \(L^2({\mathbb {R}}^n)\). We characterize geometrically the partial Gevrey-type smoothing properties of these semigroups, and we sharply describe the blow-up of the associated seminorms for short times, generalizing the hypoelliptic and the quadratic cases. As a byproduct of this study, we establish partial subelliptic estimates enjoyed by fractional Ornstein–Uhlenbeck operators on the whole space by using interpolation theory.

Estimates for eigenvalues of a class of fourth order degenerate elliptic operators with a singular potential

In this paper, we consider estimates for eigenvalues of a class of fourth order degenerate elliptic operators with a singular potential. By using the Fourier transformation and some sub-elliptic estimates, we get the Li-Yau type lower bound estimates for the eigenvalues. Especially, if the degenerate vector fields are Grushin type vector fields satisfying the generalized Métivier condition, we not only obtain lower bound estimates, but also get the Yang-type inequalities as upper bounds for eigenvalues by using the trail function.

The Gevrey smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off

In this article we study the Gevrey regularization effect for the spatially inhomogeneous Boltzmann equation without angular cutoff. This equation is partially elliptic in the velocity direction and degenerates in the spatial variable. We consider the nonlinear Cauchy problem for the fluctuation around the Maxwellian distribution and prove that any solution with mild regularity will become smooth in the Gevrey class at positive time with the Gevrey index depending on the angular singularity. Our proof relies on the symbolic calculus for the collision operator and the global subelliptic estimate for the Cauchy problem of the linearized Boltzmann operator.

Kramers–Fokker–Planck operators with homogeneous potentials

In this article we establish a global subelliptic estimate for Kramers–Fokker–Planck operators with homogeneous potentials V(q) under some conditions, involving in particular the control of the eigenvalues of the Hessian matrix of the potential. Our subelliptic lower bounds are the optimal ones up to some logarithmic correction.

Small-time asymptotics of hypoelliptic heat kernels near the diagonal, nilpotentization and related results

We establish small-time asymptotic expansions for heat kernels of hypoelliptic Hormander operators in a neighborhood of the diagonal, generalizing former results obtained in particular by Metivier and by Ben Arous. The coefficients of our expansions are identified in terms of the nilpotentization of the underlying sub-Riemannian structure. Our approach is purely analytic and relies in particular on local and global subelliptic estimates as well as on the local nature of small-time asymptotics of heat kernels. The fact that our expansions are valid not only along the diagonal but in an asymptotic neighborhood of the diagonal is the main novelty, useful in view of deriving Weyl laws for subelliptic Laplacians. In turn, we establish a number of other results on hypoelliptic heat kernels that are interesting in themselves, such as Kac's principle of not feeling the boundary, asymptotic results for singular perturbations of hypoelliptic operators, global smoothing properties for selfadjoint heat semigroups.

Algorithms for subelliptic multipliers in ℂ²

We give examples of pseudoconvex domains of finite type in C 2 \mathbb {C}^2 where the Kohn algorithm for subelliptic estimates fails to yield an effective lower bound for the order of subellipticity in terms of the type. We show how to modify the algorithm to obtain an effective procedure to prove subellipticity on domains of finite type in C 2 \mathbb {C}^2 with real analytic boundary satisfying a condition slightly stronger than pseudoconvexity. We close with a generalization to higher dimensions.

Triangular resolutions and effectiveness for holomorphic subelliptic multipliers

A solution to the effectiveness problem in Kohn's algorithm for generating holomorphic subelliptic multipliers is provided for general classes of domains of finite type in Cn, that include the so-called special domains given by finite and infinite sums of squares of absolute values of holomorphic functions. Also included is a more general class of domains recently discovered by M. Fassina [23]. More generally, for any smoothly bounded pseudoconvex domain we introduce an invariantly defined associated sheaf S of C-subalgebras of holomorphic function germs, that combined with a result of Fassina, reduces the existence of effective subelliptic estimates at p to a purely algebraic geometric question of controlling the multiplicity of S.Our main new tool, a triangular resolution, is the construction of subelliptic multipliers decomposable as Q∘Γ, where Γ is constructed from pre-multipliers and Q is part of a triangular system. The effectiveness is proved via a sequence of newly proposed procedures, called here meta-procedures, built on top of the holomorphic Kohn's procedures, where the order of subellipticity can be effectively tracked. Important sources of inspiration are algebraic geometric techniques by Y.-T. Siu [54,55] and procedures for triangular systems by D.W. Catlin and J.P. D'Angelo [16,8].The proposed procedures are purely algebraic and as such can be of wider interest for geometric and computational problems involving Jacobian determinants, such as resolving singularities of holomorphic maps.

Smoothing properties of fractional Ornstein-Uhlenbeck semigroups and null-controllability

We study fractional hypoelliptic Ornstein-Uhlenbeck operators acting on L2(Rn) satisfying the Kalman rank condition. We prove that the semigroups generated by these operators enjoy Gevrey regularizing effects. Two byproducts are derived from this smoothing property. On the one hand, we prove the null-controllability in any positive time from thick control subsets of the associated parabolic equations posed on the whole space. On the other hand, by using interpolation theory, we get global L2 subelliptic estimates for the these operators.

GLOBAL SUBELLIPTIC ESTIMATES FOR KRAMERS–FOKKER–PLANCK OPERATORS WITH SOME CLASS OF POLYNOMIALS

AbstractIn this article, we study some Kramers–Fokker–Planck operators with a polynomial potential $V(q)$ of degree greater than two having quadratic limiting behaviour. This work provides an accurate global subelliptic estimate for Kramers–Fokker–Planck operators under some conditions imposed on the potential.

Global hypoellipticity for strongly invariant operators

In this note, by analyzing the behavior at infinity of the matrix symbol of an invariant operator P with respect to a fixed elliptic operator, we obtain a necessary and sufficient condition to guarantee that P is globally hypoelliptic. As an application, we obtain the characterization of global hypoellipticity on compact Lie groups and examples on the sphere and the torus. We also investigate relations between the global hypoellipticity of P and global subelliptic estimates.

Global hypoelliptic and symbolic estimates for the linearized Boltzmann operator without angular cutoff

In this article we provide global subelliptic estimates for the linearized inhomogeneous Boltzmann equation without angular cutoff, and show that some global gain in the spatial direction is available although the corresponding operator is not elliptic in this direction. The proof is based on a multiplier method and the so-called Wick quantization, together with a careful analysis of the symbolic properties of the Weyl symbol of the Boltzmann collision operator.

Superharmonic functions associated with hypoelliptic non-Hörmander operators

In this paper, we consider a class of degenerate-elliptic linear operators [Formula: see text] in quasi-divergence form and we study the associated cone of superharmonic functions. In particular, following an abstract Potential-Theoretic approach, we prove the local integrability of any [Formula: see text]-superharmonic function and we characterize the [Formula: see text]-superharmonicity of a function [Formula: see text] in terms of the sign of the distribution [Formula: see text]; we also establish some Riesz-type decomposition theorems and we prove a Poisson–Jensen formula. The operators involved are [Formula: see text]-hypoelliptic but they do not satisfy the Hörmander Rank Condition nor subelliptic estimates or Muckenhoupt-type degeneracy conditions.

Quaternionic structure and analysis of some Kramers–Fokker–Planck operators

The present article is concerned with global subelliptic estimates for Kramers–Fokker–Planck operators with polynomials of degree less than or equal to two. The constants appearing in those estimates are accurately formulated in terms of the coeffici

Unique continuation property for multi-terms time fractional diffusion equations

A Carleman estimate and the unique continuation property of solutions for a multi-terms time fractional diffusion equation up to order $\alpha\,\,(0<\alpha<2)$ and general time dependent second order strongly elliptic time elliptic operator for the diffusion. The estimate is derived via some subelliptic estimate for an operator associated to this equation using calculus of pseudo-differential operators. A special Holmgren type transformation which is linear with respect to time is used to show the local unique continuation of solutions. We developed a new argument to derive the global unique continuation of solutions. Here the global unique continuation means as follows. If u is a solution of the multi-terms time fractional diffusion equation in a domain over the time interval $(0,T)$, then a zero set of solution over a subdomain of $\Omega$ can be continued to $(0,T)\times\Omega$.

From semigroups to subelliptic estimates for quadratic operators

Using an approach based on the techniques of FBI transforms, we give a new simple proof of the global subelliptic estimates for non-self-adjoint nonelliptic quadratic differential operators, under a natural averaging condition on the Weyl symbols of the operators, established by the second author in Subelliptic estimates for quadratic differential operators (Amer. J. Math. 133 (2011), no. 1, 39–89). The loss of the derivatives in the subelliptic estimates depends directly on algebraic properties of the Hamilton maps of the quadratic symbols. Using the FBI point of view, we also give accurate smoothing estimates of Gelfand–Shilov type for the associated heat semigroup in the limit of small times.