We have been told that the Riesz potential in $\mathbb{R}^d$, $I\_{\alpha}(f)$, defined by
$$I\_{\alpha}(f)(x):= C\int\_{\mathbb{R}^d} \frac{f(y)}{\left| x-y \right|^{d-\alpha}}\,\mathrm{d}y $$
has the following regularizing property.


For a function $f\in C^{\beta}$ (the Hölder class), we have $I\_{\alpha}(f) \in C^{\alpha+\beta}$, but we can neither prove it nor find a reference for this. 
Does it sound familiar to any of you? Do you know any reference?


Many thanks for considering my request.



Reference request: Riesz operator over Hölder class
First of all, the Riesz potential operator is not even well-defined on the Hölder class $C^\alpha$. For example, it diverges when applied to non-zero constant functions. You need to restrict to a narrower class of functions.


Second, any statement of this form requires a particular definition of $C^\alpha$ when $\alpha$ is an integer: in this case $C^\alpha$ should be defined in terms of second order differences, and it is a proper subset of the space traditionally denoted by $C^{\alpha-1,1}$.


Theorem 4 in Section V.4.4 of the classical Stein's book *Singular Integrals and Differentiability Properties of Functions* asserts that the *Bessel potential operator* $J\_\alpha$ is an isomorphism between $C^\beta$ and $C^{\beta + \alpha}$. The operator $J\_\alpha$ is a Fourier multiplier with symbol $(1 + |\xi|^2)^{-\alpha/2}$, instead of the symbol $|\xi|^{-\alpha}$ of the Riesz transform $I\_\alpha$.


If we only consider, say, compactly supported $f$, then it is easy to switch from Bessel to Riesz potential operator: simply apply Lemma 2 from Section V.3.2 of Stein's book, which asserts that the operator $J\_\alpha^{-1} I\_\alpha$ is a convolution with a finite measure, and write $I\_\alpha f = J\_\alpha ((J\_\alpha^{-1} I\_\alpha)(f))$.


You will find a lot of similar results also in Samko's book *Hypersingular Integrals and Their Applications*.


