Following the well-known theory of Beurling and Roumieu ultradistributions, we investigate new spaces of ultradistributions as dual spaces of test functions which correspond to associated functions of logarithmic-type growth at infinity. In the given framework we prove that boundary values of analytic functions with the corresponding logarithmic growth rate towards the real domain are ultradistributions. The essential condition for that purpose, condition $(M.2)$ in the classical ultradistribution theory, is replaced by the new one, $\widetilde{(M.2)}$. For that reason, new techniques were performed in the proofs. As an application, we discuss the corresponding wave front sets.
In this paper we give a global characterisation of classes of ultradifferentiable functions and corresponding ultradistributions on a compact manifold $X$. The characterisation is given in terms of the eigenfunction expansion of an elliptic operator on $X$. This extends the result for analytic functions on compact manifold by Seeley, and the characterisation of Gevrey functions and Gevrey ultradistributions on compact Lie groups and homogeneous spaces by the authors.
In this paper we analyse the structure of the spaces of coefficients of eigenfunction expansions of functions in Komatsu classes on compact manifolds, continuing the research in our previous paper. We prove that such spaces of Fourier coefficients are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on spaces of Fourier coefficients and characterise their adjoint mappings. In particular, the considered classes include spaces of analytic and Gevrey functions, as well as spaces of ultradistributions, yielding tensor representations for linear mapp…
In this paper we introduce appropriate associated function to the sequence $M_p=p^{\t p^{\s}}$, $p\in \N$, $\t>0$, $\s>1$, and derive its sharp asymptotic estimates in terms of the Lambert $W$ function. These estimates are used to prove a Paley-Wiener type theorem for compactly supported functions from extended Gevrey classes. As an application, we discuss properties of the corresponding wave front sets.
We study the regularity of smooth functions whose derivatives are dominated by sequences of the form $M_p^{τ,\s}=p^{τp^{\s}}$, $τ>0$, $\s\geq1$. We show that such functions can be characterized through the decay properties of their short-time Fourier transforms (STFT), and recover \cite[Theorem 3.1]{CNR} as the special case when $ \t>1$ and $\s = 1$, i.e. when the Gevrey type regularity is considered. These estimates lead to a Paley-Wiener type theorem for extended Gevrey classes. In contrast to the related result from \cite{PTT-05, PTT-04}, here we relax the assumption on compact support of t…
We construct differential algebras in which spaces of (one-dimensional) periodic ultradistributions are embedded. By proving a Schwartz impossibility type result, we show that our embeddings are optimal in the sense of being consistent with the pointwise multiplication of ordinary functions. In particular, we embed the space of hyperfunctions on the unit circle into a differential algebra in such a way that the multiplication of real analytic functions on the unit circle coincides with their pointwise multiplication. Furthermore, we introduce a notion of regularity in our newly defined algebra…
The main aim of this paper is to compare two recent approaches for investigating the interspace between the union of Gevrey spaces $\mathcal G_t (U)$ and the space of smooth functions $C^{\infty}(U)$. The first approach in the style of Komatsu is based on the properties of two parameter sequences $M_p=p^{τp^σ}$, $τ>0$, $σ>1$. The other one uses weight matrices defined by certain weight functions. We prove the equivalence of the corresponding spaces in the Beurling case by taking projective limits with respect to matrix parameters, while in the Roumieu case we need to consider a larger space th…
We obtain a characterization of ${\mathcal S}^{\{M_p\}}_{\{M_p\}}(\mathbb R^n)$ and $\mathcal {S}^{(M_p)}_{(M_p)}(\mathbb {R}^n)$, the general Gelfand-Shilov spaces of ultradifferentiable functions of Roumieu and Beurling type, in terms of decay estimates for the Fourier coefficients of their elements with respect to eigenfunction expansions associated to normal globally elliptic differential operators of Shubin type. Moreover, we show that the eigenfunctions of such operators are absolute Schauder bases for these spaces of ultradifferentiable functions. Our characterization extends earlier re…
An interpolation problem for extended Gevrey regularity
University of Novi Sad（诺维萨德大学）；Technical Faculty ”Mihajlo Pupin”, University of Novi Sad（诺维萨德大学米哈伊洛·皮平技术学院）；Faculty of Sciences, University of Novi Sad（诺维萨德大学理学院）
We use sequences which depend on two parameters to define families of ultradifferentiable functions which contain Gevrey classes. It is shown that such families are closed under superposition, and therefore inverse closed as well. Furthermore, we study partial differential operators whose coefficients satisfy the extended Gevrey regularity. To that aim we introduce appropriate wave front sets and derive a theorem on propagation of singularities. This extends related known results in the sense that weaker assumptions on the regularity of the coefficients are imposed.
We prove a new invariant torus theorem, for $α$-Gevrey smooth Hamiltonian systems, under an arithmetic assumption which we call the $α$-Bruno-Rüssmann condition, and which reduces to the classical Bruno-Rüssmann condition in the analytic category. Our proof is direct in the sense that, for analytic Hamiltonians, we avoid the use of complex extensions and, for non-analytic Hamiltonians, we do not use analytic approximation nor smoothing operators. Following Bessi, we also show that if a slightly weaker arithmetic condition is not satisfied, the invariant torus may be destroyed. Crucial to this …
We consider bilinear pseudo-differential operators whose symbols posses Gevrey type regularity and may have a sub-exponential growth at infinity, together with all their derivatives. It is proved that those symbol classes can be described by the means of the short-time Fourier transform and modulation spaces. Our first main result is the invariance property of the corresponding bilinear operators. Furthermore we prove the continuity of such operators when acting on modulation spaces. As a consequence, we derive their continuity on anisotropic Gelfand-Shilov type spaces. We consider both Beurli…
We define and study classes of smooth functions which are less regular than Gevrey functions. To that end we introduce two-parameter dependent sequences which do not satisfy Komatsu's condition (M.2)', which implies stability under differential operators within the spaces of ultradifferentiable functions. Our classes therefore have particular behavior under the action of differentiable operators. On a more advanced level, we study microlocal properties and prove that $${\rm WF}_{0,\infty}(P(D)u)\subseteq {\rm WF}_{0,\infty}(u)\subseteq {\rm WF}_{0,\infty}(P(D)u) \cup {\rm Char}(P),$$ where $u$…
Let A be a positive self-adjoint linear operator acting on a real Hilbert space H and $α$, c be positive constants. We show that all solutions of the evolution equation u + Au + cA $α$ u = 0 with u(0) $\in$ D(A 1 2), u (0) $\in$ H belong for all t > 0 to the Gevrey space G(A, $σ$) with $σ$ = min{ 1 $α$ , 1 1--$α$ }. This result is optimal in the sense that $σ$ can not be reduced in general. For the damped wave equation (SDW) $α$ corresponding to the case where A = --$Δ$ with domain D(A) = {w $\in$ H 1 0 ($Ω$), $Δ$w $\in$ L 2 ($Ω$)} with $Ω$ any open subset of R N and (u(0), u (0)) $\in$ H 1 0 …
We study spaces of ultradifferentiable functions which contain Gevrey classes. Although the corresponding defining sequences do not satisfy Komatsu's condition (M.2)', we prove appropriate continuity properties under the action of (ultra)differentiable operators. Furthermore, we study convenient localization procedure which leads to the concept of wave-front set with respect to our regularity conditions. As an application, we identify the standard projections of intersections/unions of wave-front sets as singular supports of suitable spaces of ultradifferentiable functions.
The overall goal of this dissertation is to investigate certain classical results from harmonic analysis, replacing the Euclidean setting, the abelian structure and the elliptic Laplace operator with a non-commutative environment and hypoelliptic operators. More specifically, we consider wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups with time-dependent non-negative propagation speeds that are Hölder-regular or even more so. The corresponding Euclidean problem has been extensively studied in the `80s and some additional results have been recently obta…
