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1.
Let L be a homogeneous left-invariant differential operator on a Carnot group. Assume that both L and Lt are hypoelliptic. We study the removable sets for L-solutions. We give precise conditions in terms of the Carnot- Caratheodory Hausdorff dimension for the removability for L-solutions under several auxiliary integrability or regularity hypotheses. In some cases, our criteria are sharp on the level of the relevant Hausdorff measure. One of the main ingredients in our proof is the use of novel local self-similar tilings in Carnot groups.  相似文献   

2.
Let P be a linear partial differential operator with coefficients in the Gevrey class Gs(Tn) where Tn is the n-dimensional torus and s?1. We prove that if P is s-globally hypoelliptic in Tn then its transposed operator tP is s-globally solvable in Tn, thus extending to the Gevrey classes the well-known analogous result in the corresponding C class.  相似文献   

3.
We consider an operator P which is a sum of squares of vector fields with analytic coefficients. The operator has a non-symplectic characteristic manifold, but the rank of the symplectic form σ is not constant on Char P. Moreover the Hamilton foliation of the non-symplectic stratum of the Poisson-Treves stratification for P consists of closed curves in a ring-shaped open set around the origin. We prove that then P is analytic hypoelliptic on that open set. And we note explicitly that the local Gevrey hypoellipticity for P is G k+1 and that this is sharp.   相似文献   

4.
Let P be a linear partial differential operator with coefficients in the Gevrey class Gs. We prove first that if P is s‐hypoelliptic then its transposed operator tP is s‐locally solvable, thus extending to the Gevrey classes the well‐known analogous result in the Cclass. We prove also that if P is s‐hypoelliptic then its null space is finite dimensional and its range is closed; this implies an index theorem for s‐hypoelliptic operators. Generalizations of these results to other classes of functions are also considered.  相似文献   

5.
A linear differential operator P(x, D) = P(x1,... x n , D1,..., D n ) = ∑αγα(x)Dα with coefficients γα(x) defined in E n is called formally almost hypoelliptic in E n if all the derivatives DνξP(x, ξ) can be estimated by P(x, ξ), and the operator P(x, D) has uniformly constant power in En. In the present paper, we prove that if P(x, D) is a formally almost hypoelliptic operator, then all solutions of equation P(x, D)u = 0, which together with some of their derivatives are square integrable with a specified exponential weight, are infinitely differentiable functions.  相似文献   

6.
In this paper, I study the microlocal hypoellipticity for a class of totally characteristic operators (1.1). My main result is as follows: Under the conditions (I), (II), if the indicial operator of (1.1) is microlocally hypoelliptic in the complement ofWF x(Pu(t,·)) for anyu(t,x)∈C b ([0,T], ℰ),t∈[0,T], λ∈ℤ, then the operator (1.1) is microlocally hypoelliptic in the variablex. Supported by the Natural Science Foundation and Young Men's Science Foundation of Academia Sinica  相似文献   

7.
In this article we study the spaces which have operator norm localization property. We prove that a finitely generated group Γ which is strongly hyperbolic with respect to a collection of finitely generated subgroups {H1,…,Hn} has operator norm localization property if and only if each Hi, i=1,2,…,n, has operator norm localization property. Furthermore we prove the following result. Let π be the fundamental group of a connected finite graph of groups with finitely generated vertex groups GP. If GP has operator norm localization property for all vertices P then π has operator norm localization property.  相似文献   

8.
Let dR be the differential of a strongly continuous representation of a Lie group G on a Hilbert space H. Let P be the left-invariant second-order differential operator on G with positive semidefinite main part P2 and with first-order part P1. If Im(P1) is in some sense subordinate to P2 then dR(P) is a pregenerator of a strongly continuous semigroup of operators in H. If the whole P1 is in some sense subordinate to P2 then that semigroup is holomorphic or, even more, dR(P) is a pregenerator of a cosine operator function.  相似文献   

9.
A linear differential operator P(D) = P(D 1, …, D n ) with constant coefficients is called almost hypoelliptic if all the derivatives D α P of the characteristic polynomial P(ξ 1, …, ξ n ) can be estimated by P. The paper proves that if P is an almost hypoelliptic operator and f is an infinitely differentiable function, square-summable with a definite exponential weight, then any square summable with the same weight solution u of the equation P(D)u = f is again an infinitely differentiable function and P(ξ) → as ξ.  相似文献   

10.
We consider pseudodifferential operators on functions on Rn+1 which commute with the Euler operator, and can thus be restricted to spaces of functions homogeneous of some given degree. The symbols of such restrictions can be regarded as functions on a reduced phase space, isomorphic to the homogeneous space Gn/Hn=SL(n+1,R)/GL(n,R), and the resulting calculus is a pseudodifferential analysis of operators acting on spaces of appropriate sections of line bundles over the projective space Pn(R): these spaces are the representation spaces of the maximal degenerate series (πiλ,ε) of Gn. This new approach to the quantization of Gn/Hn, already considered by other authors, has several advantages: as an example, it makes it possible to give a very explicit version of the continuous part from the decomposition of L2(Gn/Hn) under the quasiregular action of Gn. We also consider interesting special symbols, which arise from the consideration of the resolvents of certain infinitesimal operators of the representation πiλ,ε.  相似文献   

11.
Let H be a separable complex Hilbert space, A a von Neumann algebra in ?(H),a faithful, normal state on A. We prove that if a sequence (Xn: n ≥ 1) of uncorrelated operators in A is bundle convergent to some operator X in A and Σn=1n−2 Var(Xn) log2(n + 1) < ∞, then X is proportional to the identity operator on H. We also prove an analogous theorem for certain uncorrelated vectors in the completion L2=L2(A,φ) of A given by the Gelfand-Naimark-Segal representation theorem. Both theorems were motivated by a recent one proved by Etemadi and Lenzhen in the classical commutative setting.  相似文献   

12.
本文给出了拟齐次线性偏微分算子P(x,D)在原点为解析亚椭圆的一个必要条件:若u∈S,(Rn);且Pu=0;则存在整函数v;Pv=0,v和u在x=0的某个邻域中重合  相似文献   

13.
We obtain some general results on Sasakian Lie algebras and prove as a consequence that a (2n + 1)-dimensional nilpotent Lie group admitting left-invariant Sasakian structures is isomorphic to the real Heisenberg group H 2n + 1. Furthermore, we classify Sasakian Lie algebras of dimension five and determine which of them carries a Sasakian α-Einstein structure. We show that a five-dimensional solvable Lie group with a left-invariant Sasakian structure and which admits a compact quotient by a discrete subgroup is isomorphic to either H 5 or a semidirect product ? ? (H 3 × ?). In particular, the compact quotient is an S 1-bundle over a four-dimensional Kähler solvmanifold.  相似文献   

14.
It is proved that a polynomial (the symbol of a differential operator), the Newton polygon of which is a rectangular parallelepiped with a vertex at the origin, is almost hypoelliptic if and only if it is regular. Also some algebraic conditions of almost hypoellipticity are obtained for nonregular polynomials increasing at infinity. The results are unimprovable for polynomials of two variables.  相似文献   

15.
Let H2(S) be the Hardy space on the unit sphere S in Cn, n?2. Consider the Hankel operator Hf=(1−P)Mf|H2(S), where the symbol function f is allowed to be arbitrary in L2(S,dσ). We show that for p>2n, Hf is in the Schatten class Cp if and only if fPf belongs to the Besov space Bp. To be more precise, the “if” part of this statement is easy. The main result of the paper is the “only if” part. We also show that the membership HfC2n implies fPf=0, i.e., Hf=0.  相似文献   

16.
It is shown that if P is a linear partial differential uperator with analytic coefficients defined near a point xo in Rn and if P in Rn - 0 is such that: the principal symbol pm,(x, ξ) vanishes at (x0. ξ0). the differential of pm, with respect to ξ is different from zero at (x0, ξ0). the Poisson bracket {Pm, Pm} is zero at (x0. ξ0) and the Poisson bracket {pm, {pm.pm }} is different from zero at (x0, ξ0), then P is analytic hypoelliptic at (x0, ξ0). It is also proved that P is analytic hypoelliptic under the assumption that the first non-vanishing repeated Poisson bracket of pm, and pm, is of odd length and under some additional hypothesis on the commutators of the Hamilton fields of Re pm, and Im pm,  相似文献   

17.
A self-adjoint operator H with an eigenvalue λ embedded in the continuum spectrum is considered. Boundedness of all operators of the form AnP is proved, where P is the eigenprojection associated with λ and A is any self-adjoint operator satisfying Mourre's inequality in a neighborhood of λ and such that the higher commutators of H with A up to order n+2 are relatively bounded with respect to H.  相似文献   

18.
A polynomial P(ξ) = P(ξ1,..., ξ n ) is said to be almost hypoelliptic if all its derivatives D ν P(ξ) can be estimated from above by P(ξ) (see [16]). By a theorem of Seidenberg-Tarski it follows that for each polynomial P(ξ) satisfying the condition P(ξ) > 0 for all ξ ∈ R n , there exist numbers σ > 0 and T ∈ R1 such that P(ξ) ≥ σ(1 + |ξ|) T for all ξ ∈ R n . The greatest of numbers T satisfying this condition, denoted by ST(P), is called Seidenberg-Tarski number of polynomial P. It is known that if, in addition, P ∈ I n , that is, |P(ξ)| → ∞ as |ξ| → ∞, then T = T(P) > 0. In this paper, for a class of almost hypoelliptic polynomials of n (≥ 2) variables we find a sufficient condition for ST(P) ≥ 1. Moreover, in the case n = 2, we prove that ST(P) ≥ 1 for any almost hypoelliptic polynomial P ∈ I2.  相似文献   

19.
In this paper we give several global characterisations of the Hörmander class \(\Psi ^m(G)\) of pseudo-differential operators on compact Lie groups in terms of the representation theory of the group. The result is applied to give criteria for the ellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several examples of the first and second order globally hypoelliptic differential operators are given, in particular of operators that are locally not invertible nor hypoelliptic but globally are. Where the global hypoelliptiticy fails, one can construct explicit examples based on the analysis of the global symbols.  相似文献   

20.
This paper derives sharp L 2-coercivity inequalities for the divergence operator on bounded Lipschitz regions in ? n . They hold for fields in H(div,Ω) that are orthogonal to N(div). The optimal constants in the inequality are defined by a variational principle and are identified as the least eigenvalue of a nonstandard boundary value problem for a linear biharmonic type operator. The dependence of the optimal constant under dilations of the region is described and a generalization that involves weighted surface integrals is also proved. When n = 2, this also yields a similar coercivity result for the curl operator.  相似文献   

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