Injectivity of an integral operator over a bounded (hypercubic) domain - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T23:34:32Z http://mathoverflow.net/feeds/question/56280 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/56280/injectivity-of-an-integral-operator-over-a-bounded-hypercubic-domain Injectivity of an integral operator over a bounded (hypercubic) domain Ian Novak 2011-02-22T13:22:06Z 2011-02-22T13:22:06Z <p>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 <code>$$T[f](x) = \int_\Omega |x-y|^s f(y) \mathrm{d} y$$</code> for $x\in\Omega$. Is the operator injective, i.e., $T[f] = 0$ in $\Omega$ implies $f=0$ a.e. in $\Omega$?</p> <p>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 $s$)</p> <p>Any reference to related books/papers would be highly appreciated. Many thanks, Ian.</p>