Ah right, that is an issue. In any case I have probably solved what I needed to, but I cannot interfere with the question at present so I'll just accept this.

Thank you for your response, I will look up Besov spaces and Samko's book. There should be something in this regard over there.

To be more precise on how the reasoning was used, the Poisson kernel (of a domain) is some form of a convolution between the jump kernel and the Green function of the domain, as mentioned in equation (2.3). Assuming regularity of the jump kernel/Levy measure (assumption of Theorem 1.7), we obtain regularity of the Poisson kernel and therefore of any harmonic function (which is a convolution of the Poisson kernel with the initial condition).

Just curious, but a result like this should admit generalizations. For example, if $f$ and $g$ lie in two regularity classes, then $f * g$ should lie in their "sum". I saw this reasoning being used, for example, in your paper with Prof. Grzywny. (Theorem 1.7, "Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Levy processes"). Are you aware of such generalizations, and is the current result "tight" in that we cannot (in general) expect $h \in C^{\alpha+\beta+\epsilon}$ for some $\epsilon>0$?

I see what you mean, thank you very much! I think the idea of adjoining the "maxima" functions to retain the countability of the dense subset is really smart, although I'll have to test the rest out more carefully.