Suppose to define parabolic Hölder space $C^{1,2,\alpha}(\Omega)$ (where $\Omega:= (0,T)\times\mathbb{R}^n$) the space of $C^{1,2}(\Omega)$ (derivable 1 in time and 2 in space) functions such that the following norm is finite:
$$H_{\frac{\alpha}{2},\alpha,\Omega}(u):=\sup_{(t,x)\neq(s,y)\in\Omega}\dfrac{|u(t,x)-u(s,y)|}{(\|x-y\|_e+|t-s|^{\frac{1}{2}})^\alpha} $$
$$\|u\|_{C^{1,2,\alpha}(\Omega)}:=\|u\|_\infty+\|\partial_tu \|_\infty +H_{\frac{\alpha}{2},\alpha,\Omega}(\partial_tu) +\sum_{i=1}^n\|\partial_{x_i}u\|_\infty + \sum_{i,j=1}^n \|\partial^2_{x_ix_j}u\|_\infty+H_{\frac{\alpha}{2},\alpha,\Omega}(\partial_{x_ix_j}u) $$
My question is, is it true that, if $\|u\|_{C^{1,2,\alpha}(\Omega)}<\infty$ then for all $i=1,\ldots,n$ even $H_{\frac{\alpha}{2},\alpha,\Omega}(\partial_{x_i}u)<\infty$.
The question arises from the different definition of parabolic Hölder space that we can find in:
Krylov Lectures on Elliptic and parabolic equations in Hölder spaces,
Pascucci PDE and Martingale in option pricing.
We have two little bit different distances involved (but the norms are equivalent). The big difference is: the second one there is Hölder properties on partial derivative in space inside of definiton of norms, the first one instead uses the defintion above. So, are these two spaces equivalent?