Ultradifferential operators in the study of Gevrey solvability and regularity

In this work we present a new representation formula for ultradistributions using the so‐called ultradifferential operators. The main difference between our representation result and other works is that here we do not break the duality of Gevrey functions of other s and their ultradistributions, i.e., we locally represent an element of by an infinite order operator acting on a function of class . Our main application was in the local solvability of the differential complex associated to a locally integrable structure in a Gevrey environment.     

Gevrey local solvability in locally integrable structures

We consider a locally integrable real-analytic structure, and we investigate the local solvability in the category of Gevrey functions and ultradistributions of the complex \(\mathrm{d}^{\prime }\) naturally induced by the de Rham complex. We prove that the so-called condition \(Y(q)\) on the signature of the Levi form, for local solvability of \(\mathrm{d}^{\prime }u=f\) , is still necessary even if we take \(f\) in the classes of Gevrey functions and look for solutions \(u\) in the corresponding spaces of ultradistributions.     

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     

Local solvability for partial differential equations with multiple characteristics in mixed Gevrey-C ∞ spaces

In this paper, we consider a class of linear partial differential equations with multiple characteristics, whose principal part is elliptic in a set of variables. We assume that the subprincipal symbol has real part different from zero and that its imaginary part does not change sign. We then prove the local solvability of such a class of operators in mixed Gevrey-C ^∞ spaces, in the sense that the linear equation admits a local solution when the datum is Gevrey in some variables and only C ^∞ in the other ones.     

Global analytic and Gevrey solvability of sublaplacians under Diophantine conditions

In this paper we consider the problem of global analytic and Gevrey solvability for a class of partial differential operators on a torus in the form of squares of vector fields. We prove that global analytic and Gevrey solvability on the torus is equivalent to certain Diophantine approximation properties. Mathematics Subject Classification (2000) 35D05, 46E10, 46F05, 58J99     

Cauchy problem for parabolic systems with convolution operators in periodic spaces

For a class of periodic systems of parabolic type with pseudodifferential operators containing $\{ \vec p,\vec h\} $ -parabolic systems of partial differential equations, we study the properties of the fundamental matrices of the solutions and establish the well-posed solvability of the Cauchy problem for these systems in the spaces of generalized periodic functions of the type of Gevrey ultradistributions. For a particular subclass of systems, we describe the maximal classes of well-posed solvability of the Cauchy problem.     

Real-analytic solvability for differential complexes associated to locally integrable structures

Inspired by the work of Suzuki [12] on the concept of real-analytic solvability for first-order analytic linear partial differential operators we extend his results for the differential complexes associated to analytic locally integrable structures of corank one. We prove that such notion of solvability is related to the smooth solvability condition introduced by F. Treves [13] in 1983. In our arguments the natural extension to closed forms of the well-known Baouendi–Treves approximation formula, the so-called “Approximate Poincaré Lemma” (cf. [1], [14]), plays a key role.     

On Gevrey solvability and regularity

In this paper we study global C^∞ and Gevrey solvability for a class of sublaplacian defined on the torus T ³. We also prove Gevrey regularity for a class of solutions of certain operators that are globally C^∞ hypoelliptic in the N ‐dimensional torus (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Propagation of singularities for operators with constant coefficient hyperbolic-elliptic principal part

Summary In this paper we consider partial differential operators of the type P(x, D)= P_m(D)+Q(x, D), where the constant coefficient principal part P_m is supposed to be hyperbolic-elliptic. We study the propagation of Gevrey singularities for solutions u of the equation P(x, D) u=f, for ultradistributions f, finding exactly to which spaces of ultradistribuiions u microlocally belongs. The results are obtained by constructing a fundamental solution for P when the lower order part Q is with constant coefficients, and a parametrix otherwise.     

Geometric quantization for proper actions

We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the case of proper cocompact actions. Our invariant index is used to show that an analog of the Guillemin-Sternberg geometric quantization conjecture holds if M is symplectic with a Hamiltonian action of G that is proper and cocompact. This essentially solves a conjecture of Hochs and Landsman.     

Gevrey global solvability of non-singular real first-order differential operators

In this article we deal with Gevrey global solvability of non-singular first-order operators defined on an n-dimensional s-Gevrey manifold, s > 1. As done by Duistermaat and Hörmander in the C ^∞ framework, we show that Gevrey global solvability is equivalent the existence of a global cross section.     

Propagation des singularités Gevrey pour l’équation de Cauchy-Riemann dégénérée

In this paper we study the degenerate Cauchy-Riemann equation in Gevrey classes. We first prove the local solvability in Gevrey classes of functions and ultra-distributions. Using microlocal techniques with Fourier integral operators of infinite order and microlocal energy estimates, we prove a result of propagation of singularities along one dimensional bicharacteristics.      

A note on a generalized form of the Laplacian and of sub-Laplacians

In this note we prove a recent conjecture of Hasson [11]: we show that, for a locally integrable function u, a sufficient condition to be harmonic is that $ \lim\limits_{r\to 0^+} r^{-2}(M_{r}u-u) = 0 $ in the weak sense of distributions (M _r being the averaging operator on balls of radius r). We also extend this and other results to the setting of sub-Laplacians on Carnot groups.Investigation supported by University of Bologna. Funds for selected research topics.     

Global solvability for first order real linear partial differential operators

F. Treves, in [17], using a notion of convexity of sets with respect to operators due to B. Malgrange and a theorem of C. Harvey, characterized globally solvable linear partial differential operators on C^∞(X), for an open subset X of Rn.Let P=L+c be a linear partial differential operator with real coefficients on a C^∞ manifold X, where L is a vector field and c is a function. If L has no critical points, J. Duistermaat and L. Hörmander, in [2], proved five equivalent conditions for global solvability of P on C^∞(X).Based on Harvey-Treves's result we prove sufficient conditions for the global solvability of P on C^∞(X), in the spirit of geometrical Duistermaat-Hörmander's characterizations, when L is zero at precisely one point. For this case, additional non-resonance type conditions on the value of c at the equilibrium point are necessary.     

On local solvability of certain differential complexes

In any locally integrable structure a differential complex induced by the de Rham differential is naturally defined. We give necessary conditions, in terms of the signature of the Levi form, for its local solvability with a prescribed rate of shrinking.

     

The cauchy problem for a class of evolution systems of the parabolic type with unbounded coefficients

For a class of evolution systems of the parabolic type with unbounded coefficients, we study the properties of the fundamental solution matrices and establish the well-posed solvability of the Cauchy problem for these systems in spaces of distributions similar to Gevrey ultradistributions. For a subclass of such systems, we describe the maximal classes of well-posed solvability of the Cauchy problem.     

Cauchy problem for a class of degenerate kolmogorov-type parabolic equations with nonpositive genus

We study the properties of the fundamental solution and establish the correct solvability of the Cauchy problem for a class of degenerate Kolmogorov-type equations with { p^?,h^? } \left\{ {\overrightarrow p, \overrightarrow h } \right\} -parabolic part with respect to the main group of variables and nonpositive vector genus in the case where the solutions are infinitely differentiable functions and their initial values are generalized functions in the form of Gevrey ultradistributions.     

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.     

Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces

In this paper we give global characterisations of Gevrey–Roumieu and Gevrey–Beurling spaces of ultradifferentiable functions on compact Lie groups in terms of the representation theory of the group and the spectrum of the Laplace–Beltrami operator. Furthermore, we characterise their duals, the spaces of corresponding ultradistributions. For the latter, the proof is based on first obtaining the characterisation of their α-duals in the sense of Köthe and the theory of sequence spaces. We also give the corresponding characterisations on compact homogeneous spaces.