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:
- Is it possible to give a concise description of the family of solutions to this equation?
- 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.
- 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!