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HEAT KERNEL METHOD FOR THE LEVI-CIVITÁ EQUATION IN DISTRIBUTIONS AND HYPERFUNCTIONS

Part of: Distributions, generalized functions, distribution spaces

Published online by Cambridge University Press:  04 May 2015

JAEYOUNG CHUNG  and
PRASANNA K. SAHOO
JAEYOUNG CHUNG*
Affiliation:
Department of Mathematics, Kunsan National University, Kunsan 573-701, Korea email [email protected]
PRASANNA K. SAHOO
Affiliation:
Department of Mathematics, University of Louisville, Louisville, KY 40292, USA email [email protected]
*
✉[email protected]
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Abstract

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Let $G$ be a commutative group and $\mathbb{C}$ the field of complex numbers, $\mathbb{R}^{+}$ the set of positive real numbers and $f,g,h,k:G\times \mathbb{R}^{+}\rightarrow \mathbb{C}$. In this paper, we first consider the Levi-Civitá functional inequality

$$\begin{eqnarray}\displaystyle |f(x+y,t+s)-g(x,t)h(y,s)-k(y,s)|\leq {\rm\Phi}(t,s),\quad x,y\in G,t,s>0, & & \displaystyle \nonumber\end{eqnarray}$$
where ${\rm\Phi}:\mathbb{R}^{+}\times \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ is a symmetric decreasing function in the sense that ${\rm\Phi}(t_{2},s_{2})\leq {\rm\Phi}(t_{1},s_{1})$ for all $0<t_{1}\leq t_{2}$ and $0<s_{1}\leq s_{2}$. As an application, we solve the Hyers–Ulam stability problem of the Levi-Civitá functional equation
$$\begin{eqnarray}\displaystyle u\circ S-v\otimes w-k\circ {\rm\Pi}\in {\mathcal{D}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})\quad [\text{respectively}\;{\mathcal{A}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})] & & \displaystyle \nonumber\end{eqnarray}$$
 in the space of Gelfand hyperfunctions, where $u,v,w,k$ are Gelfand hyperfunctions, $S(x,y)=x+y,{\rm\Pi}(x,y)=y,x,y\in \mathbb{R}^{n}$, and $\circ$, $\otimes$, ${\mathcal{D}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})$ and ${\mathcal{A}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})$ denote pullback, tensor product and the spaces of bounded distributions and bounded hyperfunctions, respectively.

Keywords

additive functiondistributionfunctional inequalityheat kernel methodHyers–Ulam stabilityhyperfunctionLevi-Civitá equation

MSC classification

Primary: 39B82: Stability, separation, extension, and related topics
Secondary: 46F15: Hyperfunctions, analytic functionals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 1 , August 2015 , pp. 77 - 93
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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