Hölder continuity of a one-dimensional fractional kernel integral
= Hölder continuity of a one-dimensional fractional kernel integral
{c}
{title2=$g(x)=\int_0^1f(y)|x-y|^{-\alpha}\,dy$}
For $f\in L^2(0,1)$ and $0<\alpha<1/2$, the integral $g(x)=\int_0^1f(y)|x-y|^{-\alpha}\,dy$ is well defined for every real $x$. Scaling the squared kernel difference over the real line and using the <Cauchy-Schwarz inequality> give $|g(x+h)-g(x)|\leq C_\alpha\|f\|_2|h|^{1/2-\alpha}$.