Hi, I am interested in learning a bit more about this space. I have exhausted all the books available at my disposal, and none of them explain much of the basics for me. Here's a definition of this space.

The seminorm is $$[u] = \sup_{(x,t), (y,s) \in Q} \frac{|u(x,t) - u(y,s)|}{(|x-y|^2 + |t-s|)^{\frac{\alpha}{2}}},$$ and norm $$ \lVert{u}\rVert_{{C}^{k, \alpha}(\overline{Q})} = \sum_{i+2j \leq k} \lVert{\frac{\partial^{i+j}u}{\partial x^i \partial t^j}}\rVert_{C(\overline{Q})} + \sum_{i+2j = k} \bigg[\frac{\partial^{i+j}u}{\partial x^i \partial t^j}\bigg]. $$

Would someone please explain to me why the parabolic Holder space norm is chosen in the way that it is? For example, why aren't we interested in the quantity $u_{xt}$? Because it doesn't pop up in PDEs very often? Why only take the highest order seminorms in the norm?

Also, in the denominator of the expression for seminorms, usually we have the spatial $|x-y|$ term to a power higher than the $|t-s|$ term (eg. $|x-y|^2 + |t-s|$). Why is this?

Also, there are a number of different definitions for the norm of these spaces. Since these are norms we equip these spaces, are they somewhat equivalent? Does it really matter which one we use?


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