If $0 < \alpha < 1/2$ then a continuous function on the circle is $\operatorname{Lip}_\alpha$ if the Fourier coefficients satisfy $a_n = {\rm O}( n^{-\alpha})$. [*EDIT*: I had thought this was "iff" but a comment points out I may have misremembered; I will check in Katznelson later.] This is in Katznelson's book for instance. So the function you defined above isn't going to be H&ouml;lder continuous for any positive exponent, even though it's clearly continuous (absolutely convergent Fourier series).

Off the top of my head, I don't know of any particularly good source for the higher-dimensional stuff. There might be something in Katznelson's book, but I don't recall.