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Search: MSC category 46F ( Distributions, generalized functions, distribution spaces [See also 46T30] )

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1. CMB Online first

Chung, Jaeyoung
On an Exponential Functional Inequality and its Distributional Version
Let $G$ be a group and $\mathbb K=\mathbb C$ or $\mathbb R$. In this article, as a generalization of the result of Albert and Baker, we investigate the behavior of bounded and unbounded functions $f\colon G\to \mathbb K$ satisfying the inequality $ \Bigl|f \Bigl(\sum_{k=1}^n x_k \Bigr)-\prod_{k=1}^n f(x_k) \Bigr|\le \phi(x_2, \dots, x_n),\quad \forall\, x_1, \dots, x_n\in G, $ where $\phi\colon G^{n-1}\to [0, \infty)$. Also, as a distributional version of the above inequality we consider the stability of the functional equation \begin{equation*} u\circ S - \overbrace{u\otimes \cdots \otimes u}^{n-\text {times}}=0, \end{equation*} where $u$ is a Schwartz distribution or Gelfand hyperfunction, $\circ$ and $\otimes$ are the pullback and tensor product of distributions, respectively, and $S(x_1, \dots, x_n)=x_1+ \dots +x_n$.

Keywords:distribution, exponential functional equation, Gelfand hyperfunction, stability
Categories:46F99, 39B82

2. CMB 2011 (vol 55 pp. 673)

Aizenbud, Avraham; Gourevitch, Dmitry
Multiplicity Free Jacquet Modules
Let $F$ be a non-Archimedean local field or a finite field. Let $n$ be a natural number and $k$ be $1$ or $2$. Consider $G:=\operatorname{GL}_{n+k}(F)$ and let $M:=\operatorname{GL}_n(F) \times \operatorname{GL}_k(F)\lt G$ be a maximal Levi subgroup. Let $U\lt G$ be the corresponding unipotent subgroup and let $P=MU$ be the corresponding parabolic subgroup. Let $J:=J_M^G: \mathcal{M}(G) \to \mathcal{M}(M)$ be the Jacquet functor, i.e., the functor of coinvariants with respect to $U$. In this paper we prove that $J$ is a multiplicity free functor, i.e., $\dim \operatorname{Hom}_M(J(\pi),\rho)\leq 1$, for any irreducible representations $\pi$ of $G$ and $\rho$ of $M$. We adapt the classical method of Gelfand and Kazhdan, which proves the ``multiplicity free" property of certain representations to prove the ``multiplicity free" property of certain functors. At the end we discuss whether other Jacquet functors are multiplicity free.

Keywords:multiplicity one, Gelfand pair, invariant distribution, finite group
Categories:20G05, 20C30, 20C33, 46F10, 47A67

3. CMB 2008 (vol 51 pp. 618)

Valmorin, V.
Vanishing Theorems in Colombeau Algebras of Generalized Functions
Using a canonical linear embedding of the algebra ${\mathcal G}^{\infty}(\Omega)$ of Colombeau generalized functions in the space of $\overline{\C}$-valued $\C$-linear maps on the space ${\mathcal D}(\Omega)$ of smooth functions with compact support, we give vanishing conditions for functions and linear integral operators of class ${\mathcal G}^\infty$. These results are then applied to the zeros of holomorphic generalized functions in dimension greater than one.

Keywords:Colombeau generalized functions, linear integral operators, generalized holomorphic functions
Categories:32A60, 45P05, 46F30

4. CMB 2006 (vol 49 pp. 414)

Jiang, Liya; Jia, Houyu; Xu, Han
Commutators Estimates on Triebel--Lizorkin Spaces
In this paper, we consider the behavior of the commutators of convolution operators on the Triebel--Lizorkin spaces $\dot{F}^{s, q} _p$.

Keywords:commutators, Triebel--Lizorkin spaces, paraproduct
Categories:42B, 46F

5. CMB 2005 (vol 48 pp. 161)

Betancor, Jorge J.
Hankel Convolution Operators on Spaces of Entire Functions of Finite Order
In this paper we study Hankel transforms and Hankel convolution operators on spaces of entire functions of finite order and their duals.

Keywords:Hankel transform, convolution, entire functions, finite order
Category:46F12

6. CMB 2001 (vol 44 pp. 105)

Pilipović, Stevan
Convolution Equation in $\mathcal{S}^{\prime\ast}$---Propagation of Singularities
The singular spectrum of $u$ in a convolution equation $\mu * u = f$, where $\mu$ and $f$ are tempered ultradistributions of Beurling or Roumieau type is estimated by $$ SS u \subset (\mathbf{R}^n \times \Char \mu) \cup SS f. $$ The same is done for $SS_{*}u$.

Categories:32A40, 46F15, 58G07

7. CMB 1999 (vol 42 pp. 344)


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