= Solution
By the <Cauchy-Schwarz inequality>, the integral is bounded by $\|f\|_2\|k_x\|_2$, where $k_x(y)=|x-y|^{-\alpha}$. For $0\leq x\leq1$,
$$
\int_0^1|x-y|^{-2\alpha}\,dy=\frac{x^{1-2\alpha}+(1-x)^{1-2\alpha}}{1-2\alpha}<\infty.
$$
For $x$ outside the interval, the integral is bounded by the value at the nearest endpoint; the endpoint value is finite as well. Since $2\alpha<1$, the kernel is square-integrable for every real $x$, proving \b[<absolute convergence> for every $x$].
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