Gevrey hypoellipticity and solvability on the multidimensional torus of some classes of linear partial differential operators

Abstract In this paper we consider the problem of global analytic and Gevrey hypoellipticity and solvability for a class of partial differential operators on a torus. We prove that global analytic and Gevrey hypoellipticity and solvability on the torus is equivalent to certain Diophantine approximation properties. Keywords: Global hypoellipticity, Global solvability, Gevrey classes, Diophantine approximation property Mathematics Subject Classification (2000): 35D05, 46E10, 46F05, 58J99     

Global s-solvability, global s-hypoellipticity and Diophantine phenomena

In this paper we consider the problem of global Gevrey solvability for a class of sublaplacians on a toruswith coefficients in the Gevrey class Gs(TN). For this class of operators we show that global Gevrey solvability and global Gevrey hypoellipticity are both equivalent to the condition that the coefficients satisfy a Diophantine condition.     

Intermediate optimal gevrey exponents occur

Hypoellipticity in Gevrey classes G^s is characterized for a simple family of sums of suares of verctor fields satisfying the bracket hypothesis, with analytic coefficients. It is shown that hypoellipticity holds if and only if s is greater than or equal to an optimal exponent that may take on any rational value.     

Gevrey vectors in involutive tube structures and Gevrey regularity for the solutions of certain classes of semilinear systems

In this work, we introduce the notion of s-Gevrey vectors in locally integrable structures of tube type. Under the hypothesis of analytic hypoellipticity, we study the Gevrey regularity of such vectors and also show how this notion can be applied to the study of the Gevrey regularity of solutions of certain classes of semilinear equations.     

On global analytic and Gevrey hypoellipticity on the torus and the Métivier inequality

We obtain a global version in the N-dimensional torus of the Métivier inequality for analytic and Gevrey hypoellipticity, and based on it we introduce a class of globally analytic hypoelliptic operators which remain so after suitable lower order perturbations. We also introduce a new class of analytic (pseudodifferential) operators on the torus whose calculus allows us to study the corresponding perturbation problem in a far more general context.     

Analyticity of solutions of semi-linear equations with double characteristics

We prove the analyticity and Gevrey regularity of solutions of elliptic degenerate semi-linear differential equations principle part of which is a linear operator with double characteristics considered first by Gilioli and Treves. A new elementary proof for hypoellipticity in the weak sense is given.     

A class of sums of squares with a given Poisson-Treves stratification

We study a class of sum of squares exhibiting the same Poisson-Treves stratification as the Oleinik-Radkevič operator. We find three types of operators having distinct microlocal structures. For one of these we prove a Gevrey hypoellipticity theorem analogous to our recent result for the corresponding Oleinik-Radkevič operator.     

Regularity and solvability of linear differential operators in Gevrey spaces

We are interested in the following question: when regularity properties of a linear differential operator imply solvability of its transpose in the sense of Gevrey ultradistributions? This question is studied for a class of abstract operators that contains the usual differential operators with real‐analytic coefficients. We obtain a new proof of a global result on compact manifolds (global Gevrey hypoellipticity implying global solvability of the transpose), as well as some results in the non‐compact case by means of the so‐called property of non‐confinement of singularities. We provide applications to Hörmander operators, to operators of constant strength and to locally integrable systems of vector fields. We also analyze a conjecture stated in a recent paper of Malaspina and Nicola, which asserts that, in differential complexes naturally arising from locally integrable structures, local solvability in the sense of ultradistributions implies local solvability in the sense of distributions. We establish the validity of the conjecture when the cotangent structure bundle is spanned by the differential of a single first integral.     

Gevrey Hypoellipticity for Maximally Hypoelliptic Fourth Order Partial Differential Operators

Local C∞ and (sharp) Gevrey hypoellipticity are proved for maximally hypoelliptic fourth order partial differential operators of the form P = D|c|<4 aa(x)Xa in R" where the independent real vector fields Xj,j<n - l, satisfy the Hörmander condition that at most k brackets of the Xj spans the tangent space at each point.     

Analytic Hypoellipticity at Non-Symplectic Poisson-Treves Strata for Certain Sums of Squares of Vector Fields

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.      

Jackson’s integral of the Hurwitz zeta function

We give a Jacksonq-integral analogue of Euler’s logarithmic sine integral established in 1769 from several points of view, specifically from the one relating to the Hurwitz zeta function. Partially supported by Grant-in-Aid for Exploratory Research No. 15654003. Partially supported by Grant-in-Aid for Scientific Research (B) No. 15340003. Partially supported by Grant-in-Aid for Scientific Research (B) No. 15340012.     

Finitely generated groups of polynomial subgroup growth

We determine the structure of finitely generated residually finite groups in which the number of subgroups of each finite indexn is bounded by a fixed power ofn. To John Thompson, an inspiration to group theory, on his being awarded the Wolf Prize Partially supported by BSF and GIF grants. Partially supported by a BSF grant.     

Simultaneous reduction of a family of commuting real vector fields and global hypoellipticity

In this paper we consider a family of commuting real vector fields on then-dimensional torus and show that it can be transformed into a family of constant vector fields provided that there is one of them which its transposed is globally hypoelliptic. We apply this result to prove global hypoellipticity for certain classes of sublaplacians. The author was partially supported by CNPq.     

A recurrence for linear extensions

The number e(P) of linear extensions of a finite poset P is expressed in terms of e(Q) for certain smaller posets Q. The proof is based on M. Schützengerger's concept of promotions of linear extensions.Partially supported by NSF Grant #DMS-8700995.Partially supported by NSF Grant #DMS-8401376.     

Almost everywhere convergence of the inverse spherical transform onSL(2,R)

We prove the almost everywhere convergence of the inverse spherical transform ofL _p bi-K-invariant functions on the groupSL(2,R), 4/3<p≤2. The result appears to be sharp. Partially supported by the MPI Partially supported by the CMA     

Number of open sets for a topology with a countable basis

LetT be the family of open subsets of a topological space (not necessarily Hausdorff or evenT ₀). We prove that ifT has a countable base and is not countable, thenT has cardinality at least continuum. Partially supported by the Basic Research Fund, Israeli Academy of Sciences. Publication no. 454 done 6,8/1991. Partially sponsored by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

Compactly supported box-spline wavelets

A general procedure for constructing multivariate non-tensor-product wavelets that generate an orthogonal decomposition ofL ²(R)^s,s s≥1, is described and applied to yield explicit formulas for compactly supported spline-wavelets based on the multiresolution analysis ofL ²(R)^s 1≤s≤3, generated by any box spline whose direction set constitutes a unimodular matrix. In particular, when univariate cardinal B-splines are considered, the minimally supported cardinal spline-wavelets of Chui and Wang are recovered. A refined computational scheme for the orthogonalization of spaces with compactly supported wavelets is given. A recursive approximation scheme for “truncated” decomposition sequences is developed and a sharp error bound is included. A condition on the symmetry or anti-symmetry of the wavelets is applied to yield symmetric box-spline wavelets. Partially supported by ARO Grant DAAL 03-90-G-0091 Partially supported by NSF Grant DMS 89-0-01345 Partially supported by NATO Grant CRG 900158.     

Consolidation rates for two interacting systems in the plane

Summary This paper is a sequel of a paper of Cox and Griffeath “diffusive clustering in the two dimensional voter model”. We continue our study of the voter model and coalescing random walks on the two dimensional integer lattice. Some exact asymptotics concerning the rate of clustering in the former process and the coalescence rate of the latter are derived. We use these results to prove a limit law, announced in that earlier paper, concerning the size of the largest square centered at the origin which is of solid color at a large time t. Partially supported by the National Science Foundation under Grant DMS-831080 Partially supported by the National Science Foundation under Grant DMS-841317 Partially supported by the National Science Foundation under Grant DMS-830549     

Invariant hyperbolic tori for Gevrey-smooth Hamiltonian systems under Rüssmann’s non-degeneracy condition

In this paper, we prove the persistence of hyperbolic lower dimensional invariant tori for Gevrey-smooth perturbations of partially integrable Hamiltonian systems under Riissmann＇s nondegeneracy condition by an improved KAM iteration, and the persisting invariant tori are Gevrey smooth, with the same Gevrey index as the Hamiltonian.     

Positivity and strict contractivity of functions of operators

Carathéodory class functionsf(z) are described having the property that the self-adjoint part off(A) is positive definite for every contractionA whose spectral radius is less than 1. Analogous results are obtained for bounded analytic functions in the unit disc, and for the Nevanlinna class. Applications to Markov chains are indicated.Partially supported by the US Air Force Grant AFOSR-94-0293.Partially supported by the NSF Grant DMS-9500924.