Hölder continuity of a one-dimensional fractional kernel integral ($g(x)={\int }_0^{1}f(y)|x-y{|}^{-\alpha }dy$)

For $f\in L^2(0,1)$ and $0<\alpha <1/2$, the integral $g(x)={\int }_0^{1}f(y)|x-y{|}^{-\alpha }dy$ is well defined for every real $x$. Scaling the squared kernel difference over the real line and using the Cauchy-Schwarz inequality give $|g(x+h)-g(x)|\le C_{\alpha }\Vert f{\Vert }_2|h{|}^{1/2-\alpha }$.