Let a function $f(t,x)$, which is Lipshitz continuous in $x$ and $1/2$-Holder continuous in $t$. Is there any "official" and widely accepted way to denote the class of such functions ?
I am thinking of $\mathbb{C}^{0,\frac{1}{2}}$. Any suggestions on this ?
I am not aware of any "official" notation. Maybe we can take a cue from the generalized Hölder spaces? One notation for generalized Hölder spaces is
$$ f \in C^n_\omega \iff f \in L^\infty \text{ and } \sup_{|y| \leq h}\|\Delta^n_y f\| \leq C \omega(h) $$
where $\Delta^n_y$ is the $n$-th order finite difference operator with step $y$. The usual spaces correspond to $n = 1$ and $\omega(h) = |h|^\alpha$ for some $\alpha$.
By analogy an okay notation would be writing your space as $C^1_\Omega$ where $\Omega = \Omega(\Delta x, \Delta t) = |\Delta x| + \sqrt{|\Delta t|}$ or something like that, and define your seminorm accordingly.