Classes Of Ultradifferentiable Functions Research Articles

Quantizations and Global Hypoellipticity for Pseudodifferential Operators of Infinite Order in Classes of Ultradifferentiable Functions

We study the change of quantization for a class of global pseudodifferential operators of infinite order in the setting of ultradifferentiable functions of Beurling type. The composition of different quantizations as well as the transpose of a quantization are also analysed, with applications to the Weyl calculus. We also compare global \(\omega \)-hypoellipticity and global \(\omega \)-regularity of these classes of pseudodifferential operators.

On the Fourier-Laplace transform of functionals on a space of infinitely differentiable functions on a convex compact

Classes of ultradifferentiable functions are classically defined by imposing growth conditions on the derivatives of the functions. Following this approach we consider a Fréchet-Schwartz space of infinitely differentiable functions on a closure of a bounded convex domain of multidimensional real space with uniform bounds on their partial derivatives. Our aim is to obtain Paley-Wiener-Schwartz type theorem connecting properties of linear continuous functionals on this space with the behaviour of their Fourier-Laplace transforms. Very similar problems were considered by M. Neymark, B.A. Taylor, M. Langenbruch, A.V. Abanin.

Global pseudodifferential operators of infinite order in classes of ultradifferentiable functions

We develop a theory of pseudodifferential operators of infinite order for the global classes $\mathcal{S}_{\omega}$ of ultradifferentiable functions in the sense of Bj\"orck, following the previous ideas given by Prangoski for ultradifferentiable classes in the sense of Komatsu. We study the composition and the transpose of such operators with symbolic calculus and provide several examples.

A simple proof of Kotake–Narasimhan theorem in some classes of ultradifferentiable functions

We give a simple proof of a general theorem of Kotake-Narasimhan for elliptic operators in the setting of ultradifferentiable functions in the sense of Braun, Meise and Taylor. We follow the ideas of Komatsu. Based on an example of M\'etivier, we also show that the ellipticity is a necessary condition for the theorem to be true. The present new version of the paper modifies the proof of Theorem 1.4 for an observation by Hoepfner and Rampazo who pointed out that an induction hypothesis depends on a constant $C_q$ that changes in the induction process, and hence the argument might not work as it was written. However, the statement of the result was originally correct and modifying $C_q$ with a more concrete expression in the induction hypothesis, the induction procedure is easily clarified with almost the same proof. Moreover, we eliminate the condition that the weight is identically zero in the interval [0,1], showing that the statements hold true with very similar arguments.

On a class of ultradifferentiable functions

On a class of ultradifferentiable functions

A Paley–Wiener type theorem for generalized non-quasianalytic classes

Let $P$ be a hypoelliptic polynomial. We consider classes of ultradifferentiable functions with respect to the iterates of the partial differential operator $P(D)$ and prove that such classes satisfy a Paley–Wiener type theorem. These classes and th

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.

Characterizing the Phragmén-Lindelöf condition for evolution on algebraic curves

For an algebraic curve V in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb C}^k \times {\mathbb C}^n$\end{document} it is investigated when it satisfies the Phragmén-Lindelöf condition PL(ω) of evolution in certain classes of ultradifferentiable functions. Necessary as well as sufficient conditions are obtained which lead to a complete characterization for curves in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb C}\times {\mathbb C}^n$\end{document}. Some examples illustrate how these results can be applied. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Iterates and Hypoellipticity of Partial Differential Operators on Non-Quasianalytic Classes

Let P be a linear partial differential operator with constant coefficients. For a weight function ω and an open subset Ω of \({\mathbb{R}^N}\) , the class \({\mathcal{E}_{P,\{\omega\}}(\Omega)}\) of Roumieu type involving the successive iterates of the operator P is considered. The completeness of this space is characterized in terms of the hypoellipticity of P. Results of Komatsu and Newberger-Zielezny are extended. Moreover, for weights ω satisfying a certain growth condition, this class coincides with a class of ultradifferentiable functions if and only if P is elliptic. These results remain true in the Beurling case \({\mathcal{E}_{P,(\omega)}(\Omega)}\).

A comparison of two different ways to define classes of ultradifferentiable functions

We characterize the weight sequences $(M_p)_p$ such that the class of ultra-differentiable functions ${\mathcal E}_{(M_p)}$ defined by imposing conditions on the derivatives of the function in terms of this sequence coincides with a class of ultradifferentiable functions ${\mathcal E}_{(\omega)}$ defined by imposing conditions on the Fourier Laplace transform. As a corollary, we characterize the weight functions $\omega$ for which there exists a weight sequence $(M_p)_p$ such that the classes ${\mathcal E}_{(\omega)}$ and ${\mathcal E}_{(M_p)}$ coincide. These characterizations also hold in the Roumieu case. Our main results are illustrated by several examples.

Stability Zones in Whitney's Extension Problem for Ultradifferentiable Functions

We establish a connection between the growth rate of weight functions \(\omega \) generating nonquasianalytic classes of ultradifferentiable functions of Beurling and Roumieu type and the validity of an analog of Whitney's extension theorem for these classes.

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 (Mp). We show that evolution is equivalent to the validity of a Phragmen-Lindelof 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.

Analytic Extension of Smooth Functions

Let F be a closed proper subset of ℝn and let ℰ* be a class of ultradifferentiable functions. We give a new proof for the following result of Schmets and Valdivia on analytic modification of smooth functions: for every function ƒ ∈ ℰ* (ℝn) there is \({\widetilde f} \in {\cal E}_{*}(\rm R ^{n})\)which is real analytic on ℝnF and such that ∂a ƒ ¦F = ∂a ƒ ¦ F for any a ∈ ℕ0n. For bounded ultradifferentiable functions ƒ we can obtain \({\widetilde f}\)by means of a continuous linear operator.

Weight functions for classes of ultradifferentiable functions

Weight functions for classes of ultradifferentiable functions

Existence of Fundamental Solutions and Surjectivity of Convolution Operators on Classes of Ultra-Differentiable Functions

Existence of Fundamental Solutions and Surjectivity of Convolution Operators on Classes of Ultra-Differentiable Functions

Quasi-analyticity for hypoelliptic operators

Let MP, p = 0, 1, 2 ,..., be a sequence of positive numbers satisfying the conditions (2.1)-(2.3). Let Sz be an open set in R”, and let 9((M,), s2) and 9( { M,}, Q), Sz z [w”, denote the test function spaces of Beurling type and Roumieu type of ultradifferentiable functions of compact support, respectively. The corresponding ultradistribution spaces, denoted by 9’( (M,), a) and 9’( (M,}, 51) are generalizations of the Schwartz space 9’(G). Other spaces of the above type, consisting of ultradifferentiable functions on g, are I((M,), Sz) and &‘( {M,}, Sz), respectively. These are generalizations of the Gevrey-space rd(C2). For various properties of these spaces we may refer to [S, 10, 12, 163. If any result is true for both types of spaces then (M,) and (M,} will be replaced by the same MP. Assume that P(D) is a partial differential operator with constant coefficients. An ultradistribution EE g’(M,, R”) is said to be a fundamental solution of the operator P(D) if P(D) E = 6. In terms of this definition of fundamental solutions we define M,,-hypoelliptic operators, extending the definition of Schwartz of hypoelliptic operators. Nevertheless, we also extend the Schwartz criterion of hypoellipticity to M,-hypoellipticity. Let M(p) be the associated function of the sequence MP in the sense of Komatsu [ 111. Let P(c) be the hypoelliptic polynomial associated with the operator P(D). Then we extend Hbrmander’s algebraic characterization of hypoelliptic operators to M,-hypoelliptic operators in terms of the function M(p). This problem has been investigated earlier by Bjorck [6] for Beurling class of non-quasi-analytic functions, but our results are valid for quasianalytic class also. Moreover we unify both Beurling and Roumieu classes of ultradifferentiable functions. Assume that P(D) is a d-hypoelliptic operator of type p [3]. Let o denote a plane piece of the boundary of Sz, and let Qi(D),..., Q,(D) be 22 0022-0396/85 $3.00