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

Abstract

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.