I meet this problem in a unpublished paper. Consider integral equation $$ f(x)=g(A(||g||)x) $$ where $f\in H^2(\mathbb R^n)$ is a given function, $A:\mathbb R^+\rightarrow \mathbb R^+$ is a continue function, and $||\cdot||$ is $L^2$-norm. How to know whether there is a $g\in H^2(\mathbb R^n)$ meet the above equation ?
In fact, my question is harder, the integral equation is $$ f(x)=g(A(||\nabla g||)x). $$ Since I want to deal the easy situation first, I ask the above equation. Besides, I think it is not solvability for all continue $A$. There should be some request for the $A$. But I don't know what is it.
Thanks for any hint or answer.
Let's go for the harder question. Once $g$ is fixed, $A(\|\nabla g\|)$ is a constant. This suggests to look for $g(x)=f(x/a)$ for some $a>0$. From $$ g(A(\|\nabla g\|)\,x)=g\Bigl(A(\|\nabla g\|\,\frac{x}{a}\Bigr)=f(x) $$ we see that $a=A(\|\nabla g\|)$. IS this possible? We have $$ \|\nabla g\|=a^{\tfrac{n-2}{2}}\,\|\nabla f\|=A(\|\nabla g\|)^{\tfrac{n-2}{2}}\,\|\nabla f\|. $$ We impose now the condition that the equation $b=A(b)^{\tfrac{n-2}{2}}\,\|\nabla f\|$ has a solution $b>0$ and take $a=A(b)$.