## Injectivity of an integral operator over a bounded (hypercubic) domain

Let $s\in (0,2]$ and $\Omega$ be a hypercube in $\mathbb{R}^d$, i.e., tensor product of finite intervals. Consider the integral operator $$T[f](x) = \int_\Omega |x-y|^s f(y) \mathrm{d} y$$ for $x\in\Omega$. Is the operator injective, i.e., $T[f] = 0$ in $\Omega$ implies $f=0$ a.e. in $\Omega$?

Moreover, if $T[f] = g$, is there a way to describe the smoothness properties of $f$ based on the known smoothness properties of $g$? (in dependence of the value of $s$)

Any reference to related books/papers would be highly appreciated. Many thanks, Ian.

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