If $0 < \alpha < 1/2$ then a continuous function on the circle is $Lip_\alpha$ if and only if the Fourier coefficients satisfy $a_n = O( n^{-\alpha})$. This is in Katznelson's book for instance. So the function you defined above isn't going to be Hölder 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.
