TITLE: Regularity properties of convolution
QUESTION [11 upvotes]: 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?

REPLY [4 votes]: While the proof given by Kore-N is very nice and easy to extend to more general scales of function spaces, I thought someone may find it useful to have a more direct argument. So here it is.

Write $\Delta_h f(x) = f(x + h) - f(x)$. Suppose that $f \in C^\alpha$ and $g \in C^\beta$, and one of them — say, $g$ — is compactly supported. Since
$$ \Delta_h^2 (f * g)(x) = (\Delta_h f) * (\Delta_h g) ,$$
we have
$$ |\Delta_h^2 (f * g)(x)| \leqslant \|\Delta_h f\|_\infty \|\Delta_h g\|_1 \leqslant C |h|^\alpha |h|^\beta ,$$
where $C$ depends on the Hölder constants of $f$ and $g$ and the size of the support of $g$.
It is now a well-known fact that the above condition implies that $f * g$ is in the Hölder–Zygmund space $\Lambda^{\alpha + \beta}$, which:

coincides with the space of Hölder continuous functions when $\alpha + \beta < 1$;
coincides with the space traditionally denoted by $C^{1,\alpha+\beta-1}$ if $1 < \alpha + \beta < 2$;
is strictly larger than the class of Lipschitz-continuous functions, but it is reasonably close when $\alpha + \beta = 1$.

I am not aware of any simple proof of this fact. A very nice and reasonably elementary argument is given by Stein in his Singular Integrals and Differentiability Properties of Functions book (see Proposition V.8 there).

Remarks:

More generally, we may assume that $p, q, r \in [1, \infty]$ are such that $\frac{1}{p} + \frac{1}{q} = 1 + \frac{1}{r}$ (as in Young's inequality), and $\|\Delta_h f\|_p \leqslant C |h|^\alpha$ and $\|\Delta_h g\|_q \leqslant C |h|^\beta$. Then $\|\Delta_h^2 (f * g)\|_r \leqslant C' |h|^{\alpha + \beta}$.
The above argument clearly carries over to arbitrary locally compact Abelian groups.
Yet another approach uses the fact that Hölder–Zygmund spaces are interpolation spaces between the usual classes $C^k$, and proceeds as in the standard proof of Young's inequality by real interpolation.