An integral equation Abel and L operator?

[jarvisyang]

an integral equation Abel and L operator??

1. The Abel operator
The general Abel integral equation
$$\begin{gathered}\intop_{x}^{a}\dfrac{F(y)dy}{\left(y^{2}-x^{2}\right)^{\frac{1+u}{2}}}=f(x)\end{gathered}$$
has the solution
$$\begin{gathered}F(r)=-\dfrac{2\cos\frac{\pi u}{2}}{\pi}\dfrac{d}{dr}\intop_{r}^{a}\dfrac{f(x)xdx}{\left(x^{2}-r^{2}\right)^{\frac{1-u}{2}}}\end{gathered}$$
where $\intop_{x}^{a}\frac{\left(\bullet\right)dy}{\left(y^{2}-x^{2}\right)^{\frac{1+u}{2}}}$ is the well-known Abel operator
2. The $\mathcal{L}$ operator
The notation $\lambda(k,\phi-\phi_{0})$ is defined as follows:
$$\begin{gathered}\lambda(k,\phi-\phi_{0}):=\dfrac{1-k^{2}}{1-2k\cos(\phi-\phi_{0})+k^{2}}\end{gathered}$$
The $\mathcal{L}$ operator is defined as
$$\mathcal{L}(k)f(\phi)=\dfrac{1}{2\pi}\intop_{0}^{2\pi}\lambda(k,\phi-\phi_{0})f(\phi_{0})d\phi_{0}$$
Obviously, the following two properties for $\mathcal{L}$ operator are valid
$$\begin{gathered}\mathcal{L}(k_{1})\mathcal{L}(k_{2})=\mathcal{L}(k_{1}k_{2})\\ \lim_{k\rightarrow1}\mathcal{L}(k)f=f \end{gathered}$$
3. The question
Prove the following integral equation
$$\begin{gathered}4\intop_{0}^{\rho}\dfrac{dx}{\sqrt{\rho^{2}-x^{2}}}\end{gathered} \begin{gathered}\intop_{x}^{a}\dfrac{\rho_{0}d\rho_{0}}{\sqrt{\rho_{0}^{2}-x^{2}}} \mathcal{L}\left(\dfrac{x^{2}}{\rho\rho_{0}}\right)\sigma(\rho_{0,}\phi)=v(\rho_{0,}\phi) \end{gathered}$$
has the following solution
$$\begin{gathered}\sigma(y,\phi)=\dfrac{1}{\pi^{2}}\left[\dfrac{\Phi(a,y,\phi)}{\sqrt{a^{2}-y^{2}}}-\intop_{y}^{a}\dfrac{dt}{\sqrt{t^{2}y^{2}}}\begin{gathered}\intop_{0}^{t}\dfrac{\rho d\rho}{\sqrt{t^{2}-\rho^{2}}}\mathcal{L}\left(\dfrac{\rho y}{t^{2}}\right)\Delta v(\rho_{,}\phi)\end{gathered} \right]\end{gathered}$$
where
$$\begin{gathered}\Phi(t,y,\phi):=\dfrac{1}{t}\intop_{0}^{t}\dfrac{\rho d\rho}{\sqrt{t^{2}-\rho^{2}}}\dfrac{d}{d\rho}\left[\rho\mathcal{L}\left(\dfrac{\rho y}{t^{2}}\right)\Delta v(\rho_{,}\phi)\right]\end{gathered}$$
and $\Delta$ is the two-dimensional Laplace operator in polar coordinate, i.e.
$$\begin{gathered}\Delta:=\dfrac{\partial^{2}}{\partial\rho^{2}}+\dfrac{1}{\rho^{2}}\dfrac{\partial^{2}}{\partial\phi^{2}}\end{gathered}$$
4.Hints

To prove the formulae, one may make full use of the two properties of $\mathcal{L}$ operator and the approach of integration by parts.
These are the hints in the book, but I still can not figure the final formulae out with the help of these hints. So I post it here and wait for your excellent proof.
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[jvicens]

Thank you for bringing up this interesting topic. The integral equation you are referring to is a special case of the general Abel integral equation, which is widely used in many fields of science such as physics, engineering, and mathematics. The Abel operator and the \mathcal{L} operator play important roles in solving this type of integral equations.

To prove the given formula, we will start by using the first property of the \mathcal{L} operator. We have

\begin{gathered}\mathcal{L}\left(\dfrac{x^{2}}{\rho\rho_{0}}\right)\sigma(\rho_{0},\phi)=\mathcal{L}\left(\dfrac{x^{2}}{\rho\rho_{0}}\right)\intop_{\rho_{0}}^{a}\dfrac{\rho_{0}d\rho_{0}}{\sqrt{\rho_{0}^{2}-x^{2}}}
\mathcal{L}\left(\dfrac{x^{2}}{\rho\rho_{0}}\right)\sigma(\rho_{0,}\phi)=\mathcal{L}\left(\dfrac{x^{2}}{\rho}\right)\mathcal{L}\left(\dfrac{x^{2}}{\rho_{0}}\right)\sigma(\rho_{0},\phi)\end{gathered}

Using the second property of the \mathcal{L} operator, we can simplify this expression to

\begin{gathered}\mathcal{L}\left(\dfrac{x^{2}}{\rho_{0}}\right)\sigma(\rho_{0},\phi)=\sigma(\rho_{0},\phi)\end{gathered}

Next, we will use integration by parts to simplify the given integral equation. We have

\begin{gathered}\intop_{y}^{a}\dfrac{dt}{\sqrt{t^{2}-y^{2}}}\begin{gathered}\intop_{0}^{t}\dfrac{\rho d\rho}{\sqrt{t^{2}-\rho^{2}}}\mathcal{L}\left(\dfrac{\rho y}{t^{2}}\right)\Delta v(\rho_{,}\phi)=\left[\dfrac{\rho y}{t^{2}}\mathcal{L}\left(\dfrac{\rho y}{t