A classical result from Young in 1936 says that if $f\in C^\alpha$ and $g\in C^\beta$ with $\alpha+\beta>1$ then $\int f dg$ exists as a Riemann-Stieltjes integral.

However, I am interested in the converse. Clearly if the support of $f$ is disjoint with the support of $g$ then they can have as bad of analytic properties as you'd want.

However, can we have $f\in C^\alpha$ and $f$ not locally $C^{\alpha'}$ for any $\alpha'>\alpha$, $g\in C^\beta$ and $g$ not locally $C^{\beta'}$ for any $\beta'>\beta$, with $\alpha+\beta<1$, and have the Riemann Stieltjes integral converge?