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 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 - or at least if $u$ is a solution of a some parabolic equation)? Or an easy way to see that at least $u_x$ is continuous.
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
