I have a difficulty dealing with a seemingly elementary question. Let $f \in H^m(K)$, $m \geq 1$, where $K$ is a (possibly large) bounded domain in $\mathbb R^2$. Let $\Gamma$ be a smooth codimension 1 subset (manifold) of $K$, e.g. a curve. Then usual trace estimates tell that $g$, the restriction of $f$ to $\Gamma$ belongs to $H^{m-\frac 1 2}(\Gamma)$.
Yet, consider the special case where $\Gamma=C_R=\partial D_R$ with $D_R$ being the disk included in $K$, centered at any point $(x_0, y_0)$, and of radius $R>0$. To be in position to apply trace estimates, i need to define Sobolev spaces on $C_R$: as its elements are obviously periodic functions, Fourier series appear to be very natural. Accordingly, for $g$ defined on $C_R$, and identifying (as usual) with the 1D interval $[0, 2\pi R)$, $$ g(x)=\sum_k \hat g_k \exp(ikx/R), \qquad x \in [0, 2\pi R). $$ Keeping track of $R$, the definition of $H^m(C_R)$ which follows is: $$ g \in H^m(C_R) \mbox{ if } \sum_k (1+|k/R|^2)^m |\hat g_k|^2 <+\infty. $$ This agrees with what is found in many textbooks for $R=1$.
Yet, i don't understand how trace estimates may work when restricting $f \in H^m(K)$ to $g \in H^{m-\frac 1 2}(C_R)$ defined in this way. Pick, for instance, an harmonic polynomial (belonging to $C^\infty(\mathbb R^2)$) like $$ x^2 - y^2=\Re[(r\, \exp(i \ell/r))^2]=r^2\cos 2\ell/r, \qquad \ell:=r\theta \in [0, 2\pi r), $$ it seems to that its trace $g$ on $C_R$ should belong to all the $H^m(C_R)$, $m>0$, which doesn't agree with the former definition.
What did i do wrong ? Thanks.