Let $f$ be a compactly supported $C^{\alpha}$ function (that is Holder continuous with exponent $\alpha$) and let $g$ be a compactly supported $C^\beta$ function. What can we say about Holder continuity of their convolution
$$
h(x):=\int f(z-x) g (z) dz?
$$


It is quite clear that $h$ is in class $C^{\max(\alpha,\beta)}$, but is it possible to say something better? Is it true that $h$ is in class $C^{\alpha+\beta}$? How one can prove something like this? 



Regularity properties of convolution
The statement is actually true. I guess it is possible to prove in many different ways. For the purpose of generality, and because it is the only proof that I came up with by chance, my proof will refer to Hölder-Besov spaces over a torus $\mathbb(T)$. 


Let us suppose that $f \in \mathcal{C}^{\alpha} = \mathcal{B}^{\alpha}\_{\infty , \infty}$ and $g \in \mathcal{C}^{\beta}$ with $\alpha, \beta \in \mathbb{R}$ (note: the regularity is not supposed to be positive - and you get exactly the effect you are looking for in case of differentiability. The only problems appear if the regularities are integer: $\mathcal{C}^{1}$ is not exactly the space of continuously differentiable functions!).


Now we have a partition of the unity generated by $(\chi, \rho)$ through which we compute the norm of $f$. WLOG (up to a constant in the norm) we use a different partition for the norm of $g$: $(\chi, \psi)$ with $\psi$ supported in an annulus (larger than the one of $\rho$) such that $\psi \cdot \rho = \rho.$


Now we can start the computations: $$ ||f \* g||\_{\mathcal{B}^{\alpha + \beta}\_{\infty , \infty}} = \sup\_j 2^{(\alpha + \beta)j}||\Delta\_j f \* g||\_{\infty}$$ 


We get $$\Delta\_j f \* g = \mathcal{F}^{-1}(\rho\_j \mathcal{F}(f \* g)) = \mathcal{F}^{-1}(\rho\_j \mathcal{F}f \cdot \mathcal{F}g) =$$ $$ = \mathcal{F}^{-1}((\rho\_j \mathcal{F}f) \cdot (\psi\_j\mathcal{F}g)) = (\Delta\_j f )\*( \Delta\_j g)$$


Hence we get that:


$$||f \* g||\_{\mathcal{B}^{\alpha + \beta}\_{\infty , \infty}} = \le \sup\_j 2^{(\alpha + \beta)j}||(\Delta\_j f )\*( \Delta\_j g) ||\_{\infty} \lesssim \sup\_j 2^{(\alpha + \beta)j}||\Delta\_j f ||\_{\infty} \cdot || \Delta\_j g||\_{\infty} \le$$ $$\le ||f ||\_{\mathcal{B}^{\alpha}\_{\infty , \infty}} \cdot ||g ||\_{\mathcal{B}^{\beta}\_{\infty , \infty} }$$


