 Solution

By the Cauchy-Schwarz inequality, the integral is bounded by $\Vert f{\Vert }_2\Vert k_x{\Vert }_2$, where $k_x(y)=|x-y{|}^{-\alpha }$. For $0\le x\le 1$,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 absolute convergence for every $x$.