Some general preliminary considerations

Let $(\Omega,\mu)$ be a bounded measure space and $\EuScript{F}(\Omega)$ be a linear function space of bounded functions defined on $\Omega$ , i.e. $\EuScript{F}(\Omega)\subset\EuScript{L}^\infty(\Omega)$ . We would like to note two types of functionals from the dual space $\DSp$ , which will be used here:

Each function $g(x)\in\EuScript{L}^1(\Omega)$ defines a functional $\varphi\in\DSp$ in the following way: $$ \varphi(f)=\int\limits_{\Omega} g(x)\,f(x)\,d\mu. $$ Such functional we will call regular functional and function $g$ -- its generator.
For each $x\in\Omega$ , we will consider a functional $\delta_x\in\DSp$ defined as follows: \begin{equation}\label{dFn} \delta_x(f)=f(x). \end{equation}Since generally, we can not speak about values at the point for functions from $\EuScript(L)^\infty$ , in the following, we assume some regularity for functions from considered spaces, so that () is correctly defined.

Necessary notations and motivation

Let $(\Omega_x,\mu_x),\,(\Omega_y,\mu_y)$ be some bounded measure spaces; $\FOx,\GOy$ be some linear function spaces. Let $A:\FOx\rightarrow\GOy$ be a linear operator which has a well-defined inverse $A^{-1}:\GOy\rightarrow\FOx$ .

Consider an operator equation: \begin{equation}\label{OpEq} Af=g \end{equation}where $f\in\FOx$ is unknown and $g\in\GOy$ is given. We are interested to have an integral representation for solution of (). For this purpose we write: $$ f(x)=\delta_x(f)=\delta_x(A^{-1}(g))=[\, (A^{-1})^*\delta_x \,](g). $$

Definition of Green's function

If $\forall x\in\Omega_x$ the functional $(A^{-1})^*\delta_x$ is regular with generator $G(\cdot,y)\in\EuScript{L}^1(\Omega_y)$ , then $G$ is called Green's function of operator $A$ and solution of (

) admits the following integral representation: $$ f(x)=\int\limits_{\Omega_y}G(x,y)\,g(y)\,d\mu_y $$

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