Regularity of a class of subLaplacians on the 3-dimensional torus

In this paper we present a necessary and sufficient condition for a family of sums of squares operators to be globally hypoelliptic on a torus. This condition says that either a Diophantine condition is satisfied or there exists a point of finite type. Also, we describe the analytic and Gevrey versions of this result. The proof is based on L²-estimates and microlocal analysis.     

An unsolvable hypoelliptic differential operator

It is proved that the differential operatorD ₁ +ix ₁ D ₂ ² is hypoelliptic everywhere, but is not locally solvable in any open set which intersects the linex ₁=0. Thus, this operator is not contained in the usual classes of hypoelliptic differential operators. The proofs involve certain properties of the characteristic Cauchy problem for the backward heat operator.     

Local solvability for a class of evolution equations

Motivated by the celebrated example of Y. Kannai of a linear partial differential operator which is hypoelliptic but not locally solvable, we consider a class of evolution operators with real-analytic coefficients and study their local solvability both in L² and in the weak sense. In order to do so we are led to propose a generalization of the Nirenberg-Treves condition (ψ) which is suitable to our study.     

Spectral approximation of Wiener-Hopf operators with almost periodic generating function

The paper deals with spectral approximation of Wiener-Hopf operators acting on L^p -spaces by their

finite sections. The generating functions of the Wiener-Hopf operators are supposed to be continuous plus almost

periodic.While the usual spectra of the finite sections drastically fail to converge to the spectrum of the Wiener-Hopf

operator,it turns out that other spectral approximants, viz. the pseudospectra and the numerical ranges, do converge

perfectly.The proof requires a modified approach to the finite section method for Wiener-Hopf operators. This note

generalizes results obtained by Böttcher, Grudsky and Silbermann for the case of continuous generating

functions.     

Hypoellipticity and Local Solvability in Gevrey Classes

Let P be a linear partial differential operator with coefficients in the Gevrey class G^s. 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 C^∞class. 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.     

Perturbation of normally solvable linear relations in paracomplete spaces

In this paper we investigate the stability of the index, the nullity and the deficiency of normally solvable linear relations in paracomplete spaces under perturbation by strictly singular and T-strictly singular linear relations. This study led us to generalize some well-known results for operators and extend some results of small perturbation of normally solvable linear relations given by T. Alvarez (2012) [3].     

Multi-frame vectors for unitary systems

In this paper, the set of all complete multi-normalized tight frame vectors NF ^r (U) with multiplicity r and the set of all complete multi-frame vectors F ^r (U) with multiplicity r for a system U of unitary operators acting on a separable Hilbert space are characterized in terms of co-isometries and surjective operators in (U), the set of all operators which locally commute with U at Ψ ^r , a fixed complete wandering r-tuple for U. Then we study the linear combinations of multi-frame vectors for U and establish some conditions under which these combinations are still the same type of multi-frame vectors for U. Finally, we establish some interesting properties for multi-frame vectors when U is a unitary group. All these results have potential applications in the theory of multi-Gabor systems and multi-wavelet systems.     

Analysis of the Kohn Laplacian on quadratic CR manifolds

We study the Kohn Laplacian □_b^(q) acting on (0,q)-forms on quadratic CR manifolds. We characterize the operators □_b^(q) that are locally solvable and hypoelliptic, respectively, in terms of the signatures of the scalar components of the Levi form.     

Local solvability of partial differential operators on the Heisenberg group

In this article the following class of partial differential operators is examined for local solvability: Let P(X, Y) be a homogeneous polynomial of degree n ≥ 2 in the non-commuting variables X and Y. Suppose that the complex polynomial P(iz, 1) has distinct roots and that P(z, 0) = zⁿ. The operators which we investigate are of the form P(X, Y) where X = δ_x and Y = δ_y + xδ_w for variables (x, y, w) ∈ ?³. We find that the operators P (X, Y) are locally solvable if and only if the kernels of the ordinary differential operators P(iδ_x, ± x)* contain no Schwartz-class functions other than the zero function. The proof of this theorem involves the construction of a parametrix along with invariance properties of Heisenberg group operators and the application of Sobolev-space inequalities by Hörmander as necessary conditions for local solvability.     

Pseudodifferential Operators on Periodic Graphs

The main aim of the paper is Fredholm properties of a class of bounded linear operators acting on weighted Lebesgue spaces on an infinite metric graph Γ which is periodic with respect to the action of the group \mathbb Zⁿ{{\mathbb {Z}}^n} . The operators under consideration are distinguished by their local behavior: they act as (Fourier) pseudodifferential operators in the class OPS ⁰ on every open edge of the graph, and they can be represented as a matrix Mellin pseudodifferential operator on a neighborhood of every vertex of Γ. We apply these results to study the Fredholm property of a class of singular integral operators and of certain locally compact operators on graphs.     

Microlocal Analysis of Generalized Pullbacks of?Colombeau Functions

In distribution theory the pullback of a general distribution by a C ^∞-function is well-defined whenever the normal bundle of the C ^∞-function does not intersect the wave front set of the distribution. However, the Colombeau theory of generalized functions allows for a pullback by an arbitrary c-bounded generalized function. It has been shown in previous work that in the case of multiplication of Colombeau functions (which is a special case of a C ^∞ pullback), the generalized wave front set of the product satisfies the same inclusion relation as in the distributional case, if the factors have their wave front sets in favorable position. We prove a microlocal inclusion relation for the generalized pullback (by a c-bounded generalized map) of Colombeau functions. The proof of this result relies on a stationary phase theorem for generalized phase functions, which is given in the Appendix. Furthermore we study an example (due to Hurd and Sattinger), where the pullback function stems from the generalized characteristic flow of a partial differential equation.      

The discrete duality finite volume method for a class of quasi‐Newtonian Stokes flows

In this paper, we propose a discrete duality finite volume (DDFV) scheme for the incompressible quasi‐Newtonian Stokes equation. The DDFV method is based on the use of discrete differential operators which satisfy some duality properties analogous to their continuous counterparts in a discrete sense. The DDFV method has a great ability to handle general geometries and meshes. In addition, every component of the velocity gradient can be reconstructed directly, which makes it suitable to deal with the nonlinear terms in the quasi‐Newtonian Stokes equation. We prove that the proposed DDFV scheme is uniquely solvable and of first‐order convergence in the discrete L²‐norms for the velocity, the strain rate tensor, and the pressure, respectively. Ample numerical tests are provided to highlight the performance of the proposed DDFV scheme and to validate the theoretical error analysis, in particular on locally refined nonconforming and polygonal meshes.     

On the Prime Radical of PI-Representable Groups

The notion of PI-representable groups is introduced; these are subgroups of invertible elements of a PI-algebra over a field. It is shown that a PI-representable group has a largest locally solvable normal subgroup, and this subgroup coincides with the prime radical of the group. The prime radical of a finitely generated PI-representable group is solvable. The class of PI-representable groups is a generalization of the class of linear groups because in the groups of the former class the largest locally solvable normal subgroup can be not solvable.     

Multipliers and Modulus on Banach Algebras Related to Locally Compact Groups

LetGbe a locally compact group. In this paper we study moduli of products of elements and of multipliers of Banach algebras which are related to locally compact groups and which admit lattice structure. As a consequence, we obtain a characterization of operators onL^∞(G) which commute with convolutions whenGis amenable as discrete.     

On Periodic Locally Solvable Groups Decomposable into the Product of Two Locally Nilpotent Subgroups

We establish new results concerning various properties of a periodic locally solvable group G = A B with locally nilpotent subgroups A and B one of which is hyper-Abelian.     

A damped hyerbolic equation with critical exponent

The aim of this paper is twofold. First, we initiate a detailed study of the so-called X^s _θ spaces attached to a partial differential operator. This include localization, duality, microlocal representation, subelliptic estimates, solvability and L^p (L^q ) estimates. Secondly, we obtain some theorems on the unique continuation of solutions to semilinear second order hyperbolic equations across strongly pseudo-convex surfaces. These results are proved using some new L^p → L^q Carleman estimates, derived using the X^s _θ spaces. Our theorems cover the subcritical case; in the critical case, the problem remains open. Similar results hold for higher order partial differential operators, provided that characteristic set satisfies a curvature conditions.     

Solvability on the Heisenberg group

We solve in various spaces the linear equations L_αg = f , where L_α belongs to a class of transversally elliptic second order differential operators on the Heisenberg group with double characteristics and complex‐valued coefficients, not necessarily locally solvable. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the necessity of the solvable conditions of the typical boundary value problems for quasilinear hyperbolic systems

Some solvable conditions have been derived to ensure the existence and the uniqueness of the C^∞solution for the typical boundary problem on a local angular region for quasilinear hyperbolic systems in two variables[1]. These solvables conditions mean that, under the formulation of the typical boundary problem, the all order derivatives of the solution can be determined uniquely at the vertex. The main purpose of this paper is to show that these solvable conditions are also necessary. In other words, if these solvable conditions fail to hold, then the boundary value problem will either have no solution or have infinite number of solutions.     

On the partially microlocal hypoellipticity for a class of totally characteristic operators with characteristics of constant multiplicity

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