Find out functions of the form $g(x,y) = \int f(x,t) f(y,t) \lambda(dt)$

I am interested in the following question.

Given a symmetric function $g: \mathbb R^n \times \mathbb R^n \rightarrow \mathbb R$ or $\mathbb R_{+}^{n}\times \mathbb R_{+}^{n} \rightarrow 0$. I am interested in finding out whether $g$ can be written as the following form:

$$g(x,y) = \int f(x,t) f(y,t) \lambda(dt),$$ where $\lambda$ is some measure but not necessarily the standard Lebesgue measure.

For example, $g(x,y) = \min\{|x|,|y|\}$ can be written as the above form for $f(x,t) = \mathbb I(0<t<|x|)$. $g(x,y) = \frac{1}{|x|+|y|}$ can also be written as above for $f(x,t) = e^{-|x|t}\mathbb I(t>0)$.

I am wondering if there is any necessary or sufficient condition to describe the set of functions which satisfies the above assumption. One necessary condition is $g(x,y)$ needs to be positive symmetric definite.

Thanks a lot!!

My two cents: more a (very long) comment than an answer, but there is something I know about classes of functions that can be represented by the following integral $$\newcommand{\ffunc}{\odot} \newcommand{\sfunc}{\circledcirc} g(x,y) = \int f(x,t) \color{red}{f(t,y)} \lambda(\mathrm{d}t),\label{1}\tag{1} $$ similar to the one in the question, coinciding with it if $f$ is supposed symmetric.
A large class of integral operators of type \eqref{1}, when $\lambda$ is the Lebesgue measure, were introduced by Vito Volterra around 1910 (see references [1], [2] and [3], chapter IV, pp. 99-137) and studied by him and coauthors (notably Joseph Peres) under the name of "compositions" or "functional compositions". Volterra consider only the case $n=1$ and defines two kind of compositions: $$ \begin{align} g(x,y) &= \int\limits_x^y f(x,t) h(t,y)\, \mathrm{d}t\triangleq f\ffunc h (x,y)\quad x<y,\;\; x,y \in {\Bbb R},\label{c1}\tag{C1}\\ g(x,y) &= \int\limits_a^b f(x,t) h(t,y)\, \mathrm{d} t\triangleq f\sfunc h (x,y)\quad a<b,\;\; a,b \in \overline{\Bbb R}.\label{c2}\tag{C2}\\ \end{align} $$ Volterra and coworkers call "functional composition of the first kind" the integral operator \ref{c1} and the Functional composition of the second kind integral operator \ref{c2} (I use the symbols $\ffunc$ and $\sfunc$ since Volterra's notation is not easily manageable). Volterra's study focuses on first kind functional compositions, to the point that he's able to solve many classes of nonlinear integral equations. The following one is our main interest here, since it is similar to the one in the question and reduces to \eqref{1} when $n= 2$: $$ g(x,y) = \underbrace{f \ffunc f \ffunc \cdots \ffunc f}_{n} \triangleq f^{\ffunc n}(x,y)\quad n\in\Bbb N_+ \label{2}\tag{2} $$

Volterra's solution of \eqref{2} (see [3], chapter IV, §11, pp. 118-119).

Definition 1 (see [1], Lecture 1, p. 11, [2] §1, p. 182, [3] chapter IV, §1, p. 100) Two functions $f, g\in L^1(\Omega)$ are called permutable of the first kind if $$ f \ffunc g = g \ffunc f \quad $$ i.e. if their functional composition is commutative. An analogous definition was given by Volterra also for functional compositions of the second kind

Definition 2 (see [3] chapter IV, §7, p.110). A function $f\in L^1(\Omega)$ is said to be a function of order $\alpha\in \Bbb R_+$ if it can be written as $$ f(x,y)=\frac{(y-x)^{\alpha-1}}{\Gamma(\alpha)}\varphi(x,y) $$ where $\varphi\in C^0(\Omega)$ is bounded and such that $\varphi (x,x)\neq 0$ for all $x\in \Omega$.

Theorem. Let $g\in L^1(\Omega)$ a function of order $\beta$: if there exists a function $\theta\in L^1 (\Omega)\cup C^1(\overline\Omega)$ of order ${\beta}/{n}$ permutable with $g$ then we can solve integral equation \eqref{2} i.e. there exists precisely a function $f$ of order ${\beta}/{n}$ such that $$ f(x,y)\triangleq g^{\ffunc\frac{1}{n}}(x,y)\label{3}\tag{3} $$

Note that, as stated above, this result is proved by Volterra only for functional compositions of the first kind: he does not proves it for composition of the second kind, that would be more interesting from the question point of view as it would be possibly generalizable to the case $n>1$.

Notes

• The right side term of \eqref{3} is constructed in a fairly explicit way by Volterra: however if the order of $g$ is 1, an even more explicit solution is given in [2] §13, pp. 188-190.

• In the exposition above I tried to translate Volterra's definitions in a more modern language since his notation is a bit outdated: it is worth remember that, while he does not state this explicitly, he work mostly in the space $C^0(\Omega)$ where $\Omega\subseteq \Bbb R^2$ equipped by a structure of Banach space by the uniform norm, but in this case he assumes integrability of the functions involved in his developments, allowing them to have some integrable singularity (see the definition of the order of a function given below) so I stated them in $L^1(\Omega)$. It would also have been possible to consider $L^1_\text{loc}(\Omega)$ since Volterra considers only compact sets. However, this would have lead to a useless generalization in this context.

• The condition $\theta \in L^1 (\Omega)\cup C^1(\overline\Omega)$ should be introduced in order to deal with functions $g$ of general order: Volterra uses this hypothesis ([3] chapter IV, §11, p. 119) in order to transform a linear Volterra equation of the first kind in a readily solvable linear Volterra equation of the second kind, and thus explicitly construct $f$.

References

[1] Volterra, Vito, Theory of permutable functions, Princeton: Princeton University Press, pp. 66 (1915).

[2] Volterra, Vito, "Functions of composition", Rice Institute Pamphlet - Rice University Studies, 7, no. 4, pp. 181-254 (1920), JFM 48.1258.01.

[3] Volterra, Vito, Theory of functionals and of integral and integro-differential equations. Dover edition with a preface by Griffith C. Evans, a biography of Vito Volterra and a bibliography of his published works by Sir Edmund Whittaker. Unabridged republication of the first English translation, New York: Dover Publications, Inc. pp. 39+XVI+226 (1959), MR0100765, ZBL0086.10402.