Local Solvability Of First Order Differential Operators Near A Critical Point,Operators With Quadratic Symbols And The Heisenberg Group

We study questions of solvability for operators of the form p(x,D)+b, where p(x,ξ) is a real quadratic form and b?C. As one consequence, we obtain a necessary and sufficient condition for the local solvability of operators of the form L= near the critical point x=0, and prove the existence of tempered fundamental solutions whenever L is locally solvable.Our analysis of these operators is largely based on recent results about the solvabilitiy of left–invariant second order differential operators on the Heisenberg group and a transference principle for the Schrödinger representation.     

Dirichlet空间上Toeplitz算子与Berezin型变换相关的一些性质

本文研究了单位圆盘D 的Dirichlet 空间上Toeplitz 算子和小Hankel 算子. 利用Berezin 型变换讨论了Toeplitz 算子的不变子空间问题, 具有Berezin 型符号的Toeplitz 算子的渐进可乘性以及Toeplitz 算子的Riccati 方程的可解性. 应用Berezin 变换得到了Toeplitz 算子和小Hankel 算子可逆的充分条件. 此外, 还利用Hankel 算子和Berezin 变换刻画了算子2T_uv-T_uT_v-T_vT_u 的紧性, 其中函数u,v ∈ L^2,1.     

一类具重特征线性偏微分算子的局部可解性

运用Ｇａｒｄｉｎｇ不等式和Ｆｅｆｅｒｍａｎ－Ｐｈｏｎｇ不等式，建立了一类具实主会征的线性偏微分算子的局部可解性。     

A CLASS OF HOMOGENEOUS LEFT INVARIANT OPERATORS ON THE NILPOTENT LIE GROUP G^{d+2}

This paper is devoted to a class of homogeneous left invariant operators L\ on the nilpotent Lie group G^{d+2} of the form $L-\lambda=-\sum\limits_{j=1}^d X_j^2-i\sum\limits_{m=1}^2 \lambda _m T_m,\lambda=\lambda_1,\lambda_2)\in C^2$ where {X_1,\cdots ,X_d,T_1, T_2} is a base of left invariant vector fields on G^{d+2}. With aid of harmonic analysis on nilpotent Lie groups and the method of increment operators, for all admissible L_\lambda, subelliptic estimate and an explicit inverse axe given and the hypoellipticity and the global solvability are obtained. Also, the structure of the set of admissible points \lambda is described exhaustively.     

二步幂零李群上—类左不变微分算子亚椭圆的充要条件

本文我们用幂零李群表示的方法,给出了二步幂零李群上一类左不变微分算子是亚椭圆的充要条件．     

Global Solvability of Linear Differential Operators with Multiple Characteristics

In this paper, we first introduce the irreducible unitary representation of nilpotent Lie groups, then by using the irreducible unitary representation we construct a fundamental solution to a class of left invariant differential operators and thus obtain the global solvability of this kind of operators.     

Local Solvability for a Class of Nonhomogeneous Left Invariant Differential Operators on HR \bigotimes Rk

In this paper we discuss tbe local solvability of the following nonhomogeneous left invariant differential operators on the nilpotent Lie group H_n⊗R^K: P(X, Y, T, Z) = Σ_{|α+β|+ζ+|y|≤m|α+β|+2l=a}a_{αβly}X^αY^βT^lZ^y where X_j, Y_j (j = 1, 2, …, n), T, Z_j(j = l, 2, …, K) are bases of left invariant vector fields on H_n⊗R^K and a_{αβly} are complex constants.     

The uniqueness of a solution to the inverse Cauchy problem for a fractional differential equation in a Banach space

We consider linear fractional differential operator equations involving the Caputo derivative. The goal of this paper is to establish conditions for the unique solvability of the inverse Cauchy problem for these equations. We use properties of the Mittag-Leffler function and the calculus of sectorial operators in a Banach space. For equations with operators in a general form we obtain sufficient conditions for the unique solvability, and for equations with densely defined sectorial operators we obtain necessary and sufficient unique solvability conditions.     

On a Class of Weakly Hyperbolic Operators

The paper considers Cauchy problem in the Gevre type multianisotropic spaces. Necessary and sufficient conditions for unique solvability of this problem are obtained and the properties of operators (polynomials) that are hyperbolic with a specified weight are investigated.     

Generalized green operator of a boundary-value problem with degenerate pulse influence

We establish necessary and sufficient conditions for the solvability of inhomogeneous linear boundaryvalue problems for systems of ordinary differential equations with pulse influence in the case where the number of boundary conditions is not equal to the order of the differential system (Noetherian problems). We construct a generalized Green operator for boundary-value problems not all solutions of which can be extended from the left endpoint to the right endpoint of the interval where these solutions are constructed.     

Solvability of an inhomogeneous boundary value problem for the stationary magnetohydrodynamic equations for a viscous incompressible fluid

We study an inhomogeneous boundary value problem for the stationary magnetohydrodynamic equations for a viscous incompressible fluid corresponding to the case in which the tangential component of the magnetic field is specified on the boundary and the Dirichlet condition is posed for the velocity. We derive sufficient conditions on the input data for the global solvability of the problem and the local uniqueness of the solution.     

On the blow-up of the solution of an equation related to the Hamilton-Jacobi equation

A new model three-dimensional third-order equation of Hamilton-Jacobi type is derived. For this equation, the initial boundary-value problem in a bounded domain with smooth boundary is studied and local solvability in the strong generalized sense is proved; in addition, sufficient conditions for the blow-up in finite time and sufficient conditions for global (in time) solvability are obtained.     

Analytically invariant spectral resolvents of closed operators

The concept of analytically invariant subspace, in the framework of spectral resolvents, is extended to unbounded closed operators. One obtains necessary and sufficient conditions for a spectral resolvent to be analytically invariant. An application, concerning a strong type of spectral decomposition for closed operators with the spectra on the real line, concludes this study.     

On some classes of coefficient inverse problems for parabolic systems of equations

We examine the question on solvability in the Sobolev spaces of coefficient inverse problems for parabolic systems of equations with the overdetermination conditions on a collection of surfaces. Under certain conditions on the geometry of the domain and the boundary operators, the local solvability of the problem is proven. It is demonstrated that the conditions on the boundary operators are sharp and that, in some cases, the problem is not unconditionally solvable.     

On the blow-up of the solution of an equation with a gradient nonlinearity

We continue the study of a nonlinear third-order equation of the Hamilton-Jacobi type. For this equation, we consider an initial-boundary value problem in a bounded domain with smooth boundary and prove the local solvability in the strong generalized sense; in addition, we derive sufficient conditions for the blow-up in finite time and sufficient conditions for the time-global solvability.     

Solvability of nonlinear integral equations and surjectivity of nonlinear Markov operators

In the present paper, we consider integral equations, which are associated with nonlinear Markov operators acting on an infinite-dimensional space. The solvability of these equations is examined by investigating nonlinear Markov operators. Notions of orthogonal preserving and surjective nonlinear Markov operators defined on infinite dimension are introduced, and their relations are studied, which will be used to prove the main results. We show that orthogonal preserving nonlinear Markov operators are not necessarily satisfied surjective property (unlike finite case). Thus, sufficient conditions for the operators to be surjective are described. Using these notions and results, we prove the solvability of Hammerstein equations in terms of surjective nonlinear Markov operators.     

Global solvability on the torus for certain classes of formally self-adjoint operators

In this paper we prove a necessary and sufficient condition for global solvability on the torus for two classes of formally self-adjoint operators. For the first class of operators we prove that global solvability is equivalent to an algebraic condition involving Liouville vectors and simultaneous approximability. For the second class of operators, when the coefficients are not identically zero, an independence condition on the coefficients is shown to be necessary and sufficient for global solvability. Received: 21 June 1999 / Revised version: 8 May 2000     

Solvability of the inhomogeneous Cauchy–Riemann equation in projective weighted spaces

We establish an analog of Hörmander’s Theorem on solvability of the inhomogeneous Cauchy–Riemann equation for a space of measurable functions satisfying a system of uniform estimates. The result is formulated in terms of the weight sequence defining the space. The same conditions guarantee the weak reducibility of the corresponding space of entire functions. Basing on these results, we solve the problem of describing the multipliers in weighted spaces of entire functions with the projective and inductive-projective topological structure. Applications are obtained to convolution operators in the spaces of ultradifferentiable functions of Roumieu type.     

An inverse problem for operators of a triangular structure

An inverse problem for operators of a triangular structure is studied. An algorithm for the solution and necessary and sufficient conditions for the solvability of this problem are obtained, moreover uniqueness is proved. Applications to difference and differential operators are considered.     

Riquier–Neumann Problem for the Polyharmonic Equation in a Ball

We obtain necessary and sufficient conditions for the solvability of the Riquier–Neumann problem for the inhomogeneous polyharmonic equation in the unit ball.