I am interested in understanding the solutions $\phi$ of the following integral equation: $$0=\int\_0^1 \int\_0^1 \phi(x,y)(x-y)\thinspace dx\thinspace dy.$$ Equivalently, I am interested in understanding the kernel of the linear function $F:C^k(\mathbb{R}^2)\to\mathbb{R}$ where $F(\phi)=\int\_0^1 \int\_0^1 \phi(x,y)(x-y)\thinspace dx\thinspace dy$. (The particular function space for the domain doesn't matter especially much to me - I have been assuming that $k\geq1$, but if it is convenient to use a different domain, then that will work for my purposes as well.)


So far I know that any symmetric function is a solution, i.e. any $\phi$ satisfying $\phi(x,y)=\phi(y,x).$ However, I also know that not every solution is symmetric, as illustrated by the solution $\phi(x,y)=x+y^2$.


I also know that the kernel of $F$ is of codimension 1, and so its dimension is countably infinite.
My questions are:


1. Is it possible to give a concise description of the family of solutions to this equation?
2. Is it possible to explicitly find a basis for the kernel of $F$? I mean this in the sense of finding a countable collection of functions whose span is dense in the kernel.
3. More generally, in the (limited) literature I've read on integral equations, I have not seen sources which deal with integral equations of two variables. Could you point me to a source where I can learn more about equations of this type? My intuition is that this equation is of a simple enough form that it seems like somebody must have studied it before. Does anybody know of a source that discusses this equation?


Thank you in advance!



Solution set of integral equation/ Kernel of linear operator
I think it's easiest to work in the Hilbert space setting for this problem, i.e., to consider $F$ is a functional on the space $L^2([0,1]^2)$, where $[0,1]^2$ is endowed with the Lebesgue measure.


Let $1 \in L^2([0,1])$ denote the constant function with value $1$, and let $h \in L^2([0,1])$ be given by $h(x) = x$ for all $x \in [0,1]$.


Choose your favourite orthonormal basis $(e\_n)\_{n \in \mathbb{N}\_0}$ of $L^2([0,1])$ with the property that $e\_k$ is orthogonal to both $1$ and $h$ for each $k \ge 2$ (for instance, $(e\_n)\_{n \in \mathbb{N}\_0}$ can be chosen as a basis of *orthogonal polynomials* with respect to the Lebesgue measure on $[0,1]$).


Then $(e\_k \otimes e\_j)\_{(k,j) \in \mathbb{N}\_0^2}$ is an orthonormal basis of $L^2([0,1]^2)$, and it is easy to see that the following vectors are in $\ker F$:


* $e\_k \otimes e\_j$ whenver $j \ge 2$ or $k \ge 2$.
* $e\_0 \otimes e\_0$ and $e\_1 \otimes e\_1$.
* $\frac{1}{\sqrt{2}}\big(e\_0 \otimes e\_1 + e\_1 \otimes e\_0\big)$.


Hence, these vectors form an orthonormal basis of $\ker F$.


