On an Exponential Functional Inequality and its Distributional Version

Chung, Jaeyoung

Canadian mathematical bulletin, Tome 58 (2015) no. 1, pp. 30-43

Résumé

Let $G$ be a group and $\mathbb{K}\,=\,\mathbb{C}\,\text{or}\,\mathbb{R}$ . In this article, as a generalization of the result of Albert and Baker, we investigate the behavior of bounded and unbounded functions $f\,:\,G\,\to \,\mathbb{K}$ satisfying the inequality $$\left| f\left( \sum\limits_{k=1}^{n}{{{x}_{k}}} \right)\,-\,\underset{k=1}{\overset{n}{\mathop{\Pi }}}\,f\left( {{x}_{k}} \right) \right|\,\,\le \phi \left( {{x}_{2}},\,.\,.\,.\,,{{x}_{n}} \right),\,\,\,\,\,\,\forall {{x}_{1}},\,.\,.\,.\,,{{x}_{n}}\,\in \,G,$$ Where $\phi :\,{{G}^{n-1}}\,\to \,[0,\,\infty )$ . Also as a a distributional version of the above inequality we consider the stability of the functional equation $$u\,\circ \,S\,-\,\overbrace{u\,\otimes \,.\,.\,.\,\otimes \,u}^{n-\text{times}}\,=\,0,$$ where $u$ is a Schwartz distribution or Gelfand hyperfunction, $\circ$ and $\otimes$ are the pullback and tensor product of distributions, respectively, and $S\left( {{x}_{1}},\,.\,.\,.\,,{{x}_{n}} \right)\,=\,{{x}_{1}}\,+\,.\,.\,.\,+\,{{x}_{n}}$ .

Détail

DOI : 10.4153/CMB-2014-012-x

Mots-clés : 46F99, 39B82, distribution, exponential functional equation, Gelfand hyperfunction, stability