Suppose that a function $u:[0,1]\times[0,T]\to\mathbb{R}$ belongs to the parabolic Holder space $C^{2+\alpha,1+\alpha/2}$, for $\alpha\in(0,1)$.

What can be said about $u_x=\partial_x u$?

I am not an expert, I am able to show that $u_x\in C^{1+\alpha,\alpha/2}$, but I was told that it actually should belong to $C^{1+\alpha,1/2+\alpha/2}$ (unfortunately without any reference).

Does anyone know a good reference (or a proof) for this fact (if true)?

Analogously, if $u\in C^{2n+\alpha,n+\alpha/2}$ then $u_x$ should be in $C^{2n-1+\alpha,n-1/2+\alpha/2}$.

Thanks, c