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 $f$ nowhere locally $C^{\alpha'}$ for any $\alpha'>\alpha$, $g\in C^\beta$ and $g$ nowhere locally $C^{\beta'}$ for any $\beta'>\beta$, with $\alpha+\beta<1$, and have the Riemann Stieltjes integral converge?



Converse to Young's classical result on Riemann-Stieltjes integration
$\newcommand\al\alpha\newcommand\be\beta$Yes, of course. Take any $\al>0$ and $\be>0$ such that $\al+\be<1$. For $x\in[0,1]$, let
$$f(x):=\sum\_{j=1}^\infty 2^{-j}(x-r\_j)\_+^\al$$
and
$$g(x):=\sum\_{j=1}^\infty 2^{-j}(x-r\_j)\_+^\be,$$
where $(r\_1,r\_2,\dots)$ is an enumeration of the rational numbers in $[0,1)$ and $u\_+:=\max(0,u)$.


For any real $u,v$ one has $u\_+^\al-v\_+^\al\le(u-v)\_+^\al$ and hence for any $x,y$ in $[0,1]$
$$f(x)-f(y)\le\sum\_{j=1}^\infty 2^{-j}(x-y)\_+^\al=(x-y)\_+^\al,$$
so that $f\in C^\al[0,1]$. Also, for any $j$  

$$\liminf\_{x\downarrow r\_j}\frac{f(x)-f(r\_j)}{(x-r\_j)^\al}\ge2^{-j}>0$$
and hence for any real $\al'>\al$
$$\liminf\_{x\downarrow r\_j}\frac{f(x)-f(r\_j)}{(x-r\_j)^{\al'}}=\infty,$$
so that $f$ is nowhere $C^{\al'}$ on $[0,1]$.
Similarly, $g\in C^\be[0,1]$ but $g$ is nowhere $C^{\be'}$ on $[0,1]$ for any real $\be'>\be$.


Finally,
$$\int\_0^1 f\,dg=\sum\_{j=1}^\infty\sum\_{k=1}^\infty 2^{-j}2^{-k}\,I\_{j,k},$$
where
$$I\_{j,k}:=\int\_0^1(x-r\_j)\_+^\al\be(x-r\_k)\_+^{\be-1}\,dx
\le\int\_0^1\be(x-r\_k)\_+^{\be-1}\,dx\le1.$$
So, $\int\_0^1 f\,dg\in[0,1]\subset\mathbb R$. $\quad\Box$


