Is this kernel invertible ? $K(x,y)=\frac{e^{-\frac{xy}{x+y}}}{x+y}$

Question

Is the following Kernel invertible?

$K(x,y)=\frac{e^{-\frac{xy}{x+y}}}{x+y}, x\in[0,1],y\in [0, \infty)$

i.e. if $\int_0^1 K(x,y) f(x) dx=0 ,\forall y\in [0, \infty)$ can we conclude $f(x)=0,x\in [0,1]$?

Answer 1

This is not an answer but an angle of attack for this issue.

Why not expand the exponential and integrate term by term (uniform convergence assumed):

$$\sum_{k=0}^{\infty}(-1)^n y^n \int_0^1 \dfrac{1}{(x+y)^{n+1}}x^n f(x)dx$$

and the integrals can be interpreted, up to a sign change for $y$, as convolutions...

Something else: your kernel looks in connection with the type of integrals one finds in Hilbert's integral inequality; see for example

http://www.emis.de/journals/JIPAM/images/114_08_JIPAM/114_08.pdf

Answer 2

How about the following?

Make the following change of variables $x=\frac{1}{v}, y=\frac{-1}{z}$ then we would have

$\int_0^1 \frac{e^{\frac{-xy}{x+y}}}{x+y} f(x) dx= z \int_1^\infty \frac{e^{\frac{1}{z-v}}}{z-v} f(\frac{1}{v})(\frac{-1}{v}) dv=0$

does this help?