Space Of Tempered Distributions Research Articles

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$H^s(\mathbb{R}^n)$.

Surjectivity of Euler operators on temperate distributions

Euler operators are partial differential operators of the form P(θ) where P is a polynomial and θj=xj∂/∂xj. We show that every non-trivial Euler operator is surjective on the space of temperate distributions on Rd. This is in sharp contrast to the behaviour of such operators when acting on spaces of differentiable or analytic functions.

On the invertibility of Born–Jordan quantization

As a consequence of the Schwartz kernel Theorem, any linear continuous operator Aˆ: S(Rn)⟶S′(Rn) can be written in Weyl form in a unique way, namely it is the Weyl quantization of a unique symbol a∈S′(R2n). Hence, dequantization can always be performed, and in a unique way. Despite the importance of this topic in Quantum Mechanics and Time-frequency Analysis, the same issue for the Born–Jordan quantization seems simply unexplored, except for the case of polynomial symbols, which we also review in detail. In this paper we show that any operator Aˆ as above can be written in Born–Jordan form, although the representation is never unique if one allows general temperate distributions as symbols. Then we consider the same problem when the space of temperate distributions is replaced by the space of smooth slowly increasing functions which extend to entire function in C2n, with a growth at most exponential in the imaginary directions. We prove again the validity of such a representation, and we determine a sharp threshold for the exponential growth under which the representation is unique. We employ techniques from the theory of division of distributions.

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

Local Tauberian theorems in spaces of distributions related to cones, and their applications

In this article we introduce and study special spaces of distributions related to a given cone. These spaces occupy an intermediate position between the space of temperate distributions and the class of distributions concentrated on a cone. The properties of these spaces are investigated. In particular, we prove that they are convolution algebras. Quasi-asymptotic properties of distributions belonging to these spaces are thoroughly studied. To this end we prove several complex Tauberian and Abelian theorems in which the role of the integral transformation is played by the Laplace transformation. This transformation establishes an isomorphism between these spaces and the classes of functions holomorphic in special wedge-shaped domains. These results are applied to the study of the asymptotic behaviour of functions holomorphic in wedge-shaped domains at boundary points. A local theorem on non-compensation of singularities of holomorphic functions is proved.