Generic results in classes of ultradifferentiable functions

Let E be a Denjoy–Carleman class of ultradifferentiable functions of Beurling type on the real line that strictly contains another class F of Roumieu type. We show that the set S of functions in E that are nowhere in the class F   is large in the topological sense (it is residual), in the measure theoretic sense (it is prevalent), and that S∪{0}$S\cup \{0\}$ contains an infinite dimensional linear subspace (it is lineable). Consequences for the Gevrey classes are given. Similar results are also obtained for classes of ultradifferentiable functions defined imposing conditions on the Fourier–Laplace transform of the function.     

The overdetermined Cauchy problem in some classes of ultradifferentiable functions

In this paper we study the Cauchy problem for overdetermined systems of linear partial differential operators with constant coefficients in some spaces of ultradifferentiable functions of class (M _p ). We show that evolution is equivalent to the validity of a Phragmén-Lindel?f principle for entire and plurisubharmonic functions on some irreducible affine algebraic varieties, and make applications in different situations. We find necessary and sufficient conditions for well posedness, and relate the hyperbolicity of a given system to that of its principal part. Received: January 19, 1999?Published online: May 10, 2001     

Global regularity in ultradifferentiable classes

We study ω-regularity of the solutions of certain operators that are globally C ^∞-hypoelliptic in the N-dimensional torus. We also apply these results to prove the global ω-regularity for some classes of sublaplacians. In this way, we extend previous work in the setting of analytic and Gevrey classes. Different examples on local and global ω-hypoellipticity are also given.     

Holomorphic approximation of ultradifferentiable functions

Mathematische Annalen -     

Convolution equations for ultradifferentiable functions and ultradistributions

We characterize surjectivity of convolution operators on spaces of ultradifferentiable functions and ultradistributions of Beurling type in the spirit of Hörmander's convexity conditions. This completes results of Bonet, Galbis, and Meise. In contrast to the classical approach our proofs only use properties of ultradistributions and functional analytic tools.     

Almost holomorphic extensions of ultradifferentiable functions

For a smooth functionf on ℝ ⁿ , we construct an extensionF to ℂ ⁿ with vanishing to a high order on ℝ ⁿ and give precise estimates of how the degree of smothness is reflected in the degree of vanishing. This analysis is used to define the operator on (n,n−1) forms with singularities on ℝ ⁿ .     

Representation of solutions of convolution equations in nonquasianalytic beurling classes of ultradifferentiable functions of mean type

We consider convolution equations in nonquasianalytic Beurling spaces of ultradifferentiable functions of mean type. We obtain a representation for a particular solution to such an equation as an exponential series whose coefficients are determined by the right-hand side of the equation.     

Roots in differential rings of ultradifferentiable functions

Let {M _k } be a logconvex sequence satisfying the differentiability condition $$\sup (M_{n + 1} /M_n )^{1/n} < \infty $$ . It is shown that the Carleman class C{k! M _k } contains all C ^∞ roots of its nonflat elements, i.e., if f ∈ C{k! M _k } and α > 0, then $$f^\alpha \in C\{ k!M_k \} whenever f^\alpha \in C^\infty $$ . If {M _k } also satisfies the additional condition M _n ^1/n → ∞, then the Beurling class C(k! M _k ) is also contains all C ^∞ roots of its nonflat elements.     

Convolution equations and spaces of ultradifferentiable functions

We study spaces of “approximate solutions” to a given convolution equation. In particular we show how their quasianalyticity can be reduced to the study of a suitably constructed Cauchy problem for related spaces. We give a unicity theorem for this problem. This paper is dedicated to the author’s son, Alessandro. The author has been partially supported by Ministero P.I., and G.N.S.A.G.A. of Consiglio Nazionale delle Ricerche.     

Local operators between spaces of ultradifferentiable functions and ultradistributions

In this article we characterize certain ultradifferential operators by the condition of being local. First, we examine the continuity properties of local linear operators on spaces of ultradifferentiable functions in the sense of Beurling and of Roumieu. Next, a structure theorem for vector-valued ultradistributions with support at the origin is proved. This result leads to a representation theorem for continuous local operators from spaces of ultradifferentiable functions into various spaces of ultradistributions. In combination with the continuity results we thus obtain in many cases the desired characterization.     

On Borel’s theorem for spaces of ultradifferentiable functions of mean type

Spaces of ultradifferentiable functions of mean type defined by a weight function ω are considered. It is obtained a necessary and sufficient condition for ω under which the corresponding space admits an analog of Borel’s theorem.     

Ultrarapidly decreasing ultradifferentiable functions, Wigner distributions and density matrices

Spaces , , of ultradecreasing ultradifferentiable (or forshort, ultra-) functions, depending on a weight e^(x), are introducedin the context of quantum statistics. The corresponding coefficientspaces in the Fock basis are identified, and it is shown thatthe Hermite expansion is a tame isomorphism between these spaces.These results are used to link decay properties of density matricesto corresponding properties of the Wigner distribution.