The boundedness of a class of semiclassical Fourier integral operators on Sobolev space $H^{s}$

We introduce the relevant background information thatwill be used throughout the paper.Following that, we will go over some fundamental concepts from thetheory of a particular class of semiclassical Fourier integraloperators (symbols and phase functions), which will serve as thestarting point for our main goal. Furthermore, these integral operators turn out to be bounded on$S\left(\mathbb{R}^{n}\right)$ the space of rapidly decreasingfunctions (or Schwartz space) and its dual$S^{\prime}\left(\mathbb{R}^{n}\right)$ the space of temperatedistributions. Moreover, we will give a brief introduction about$H^s(\mathbb{R}^n)$ Sobolev space (with $s\in\mathbb{R}$).Results about the composition of semiclassical Fourier integraloperators with its $L^{2}$-adjoint are proved. These allow to obtainresults about the boundedness on the Sobolev spaces$H^s(\mathbb{R}^n)$.

H s -Boundedness of a class of Fourier Integral Operators

In this paper, we define a particular class of Fourier Integral Operators (FIO for short). These FIO turn out to be bounded on the spaces S (ℝ n ) of rapidly decreasing functions (or Schwartz space) and S′ (ℝ n ) of temperate distributions. Results about the composition of FIO with its L 2-adjoint are proved. These allow to obtain results about the continuity on the Sobolev Spaces.

Spectra and ergodic properties of multiplication and convolution operators on the space $${\mathcal S}({\mathbb R})$$

In this paper we investigate the spectra and the ergodic properties of the multiplication operators and the convolution operators acting on the Schwartz space $${\mathcal S}({\mathbb R})$$ S ( R ) of rapidly decreasing functions, i.e., operators of the form $$M_h: {\mathcal S}({\mathbb R})\rightarrow {\mathcal S}({\mathbb R})$$ M h : S ( R ) → S ( R ) , $$f \mapsto h f $$ f ↦ h f , and $$C_T:{\mathcal S}({\mathbb R})\rightarrow {\mathcal S}({\mathbb R})$$ C T : S ( R ) → S ( R ) , $$f\mapsto T\star f$$ f ↦ T ⋆ f . Precisely, we determine their spectra and characterize when those operators are power bounded and mean ergodic.

A Family of Banach Spaces Over ℝ∞

In T. L. Gill and W. W. Zachary, Functional Analysis and the Feynman Operator Calculus (Springer, New York, 2016), the topology of [Formula: see text] was replaced with a new topology and denoted by [Formula: see text]. This space was then used to construct Lebesgue measure on [Formula: see text] in a manner that is no more difficult than the same construction on [Formula: see text]. More important for us, a new class of separable Banach spaces [Formula: see text], [Formula: see text], for the HK-integrable functions, was introduced. These spaces also contain the [Formula: see text] spaces and the Schwartz space as continuous dense embeddings. This paper extends the work in T. L. Gill and W. W. Zachary, Functional Analysis and the Feynman Operator Calculus (Springer, New York, 2016) from [Formula: see text] to [Formula: see text].

Controllability of systems of linear partial differential equations

In a number of papers, the controllability theory was recently studied. But quite a few of them were devoted to control systems described by ordinary differential equations. In the case of systems described by partial differential equations, they were studied mostly for classical equations of mathematical physics. For example, in papers by G. Sklyar and L. Fardigola, controllability problems were studied for the wave equation on a half-axis. In the present paper, the complete controllability problem is studied for systems of linear partial differential equations with constant coefficients in the Schwartz space of rapidly decreasing functions. Necessary and sufficient conditions for complete controllability are obtained for these systems with distributed control of the special form: u(x,t)=e-αtu(x). To prove these conditions, other necessary and sufficient conditions obtained earlier by the author are applied (see ``Controllability of evolution partial differential equation''. Visnyk of V. N. Karasin Kharkiv National University. Ser. ``Mathematics, Applied Mathematics and Mechanics''. 2016. Vol. 83, p. 47-56). Thus, the system $$\frac{\partial w(x,t)}{\partial t} = P\left(\frac\partial{i\partial x} \right) w(x,t)+ e^{-\alpha t}u(x),\quad t\in[0,T], \ x\in\mathbb R^n, $$ is completely controllable in the Schwartz space if there exists α>0 such that $$\det\left( \int_0^T \exp\big(-t(P(s)+\alpha E)\big)\, dt\right)\neq 0,\quad s\in\mathbb R^N.$$ This condition is equivalent to the following one: there exists $\alpha>0$ such that $$\exp\big(-T(\lambda_j(s)+\alpha)\big)\neq 1 \quad \text{if}\ (\lambda_j(s)+\alpha)\neq0,\qquad s\in\mathbb R^n,\ j=\overline{1,m},$$ where $\lambda_j(s)$, $j=\overline{1,m}$, are eigenvalues of the matrix $P(s)$, $s\in\mathbb R^n$. The particular case of system (1) where $\operatorname{Re} \lambda_j(s)$, $s\in\mathbb R$, $j=\overline{1,m}$, are bounded above or below is studied. These systems are completely controllable. For instance, if the Petrovsky well-posedness condition holds for system (1), then it is completely controllable. Conditions for the existence of a system of the form (1) which is not completely controllable are also obtained. An example of a such kind system is given. However, if a control of the considered form does not exists, then a control of other form solving complete controllability problem may exist. An example illustrating this effect is also given in the paper.