Help in finding a function in $L^p$ such that $f*g=||g||_1 f$ with $g\in L^1(\mathbb{R})$ non negative and fixed

Question

I consider a non negative function $g\in L^1(\mathbb{R})$.

I want to find a function $f\in L^p(\mathbb{R})$ such that $$ f*g=||g||_1 \:f ,$$ where $*$ is the convolution.

I would be very thankful if someone could help me finding this function $f$.

Answer 1

Albeit not too interesting, $f \equiv 0$ works.

If the space is a probability space, $p \geq 1\,^\dagger$ and $ \|g\|_1 \neq 0$, then $h:= \frac{g}{\|g\|_1}$ is a probability density (let's say of a rnd. var. $X$) and your equation becomes: $$f * h = f$$ Now if $f$ is a probabilty density of $Y$, then $f*h$ is the PDF of $X+Y$. And the above equation says that $X+Y = Y$ with a probability of $1$. This means that $X = 0$ with a probablity of $1$. So indeed $f=0$ a.e.

$^\dagger$ see stochasticboy321's comment below.