No nontrivial estimates, at least if the definitions are right.
Say let's define $$\ell_\alpha c=\sup\,\left\{\,\sum_{t_0<\dots<t_n}|c(t_i)-c(t_{i-1}|^\alpha\,\right\}$$ and $$\dim c=\sup\{\,\alpha\mid \ell_\alpha c<\infty\,\}.$$
Then from Pythagorean theorem we get $$\dim c=\max\{\dim x, \dim y\}.$$