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If $f:\mathbb{R}_+\to\mathbb{R}_+$ is a positive function which is locally $L^1$ with $\int_0^t{f(s)ds}=at^n$ for all $t\geq0$ then, by differentiating, it follows that $f(t)=a$ if $n=1$ and $f=0=a$ otherwise.

I would like to show that, if $f:\mathbb{R}_+^n\to\mathbb{R}_+$ is a symmetric function which satisfies $$\int_0^t\int_0^{t_1}\cdots \int_0^{t_{n-1}} f(t_1,t_2,\ldots,t_n)dt_n\cdots dt_2dt_1=at^m \quad \text{for all } t\geq0$$ then we must have $$f ~\text{constant if }n=m \quad \text{and} \quad f=0~\text{ if }n\neq m.$$ Does anyone know if this is possible? Perhaps there is even a counterexample?!

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