# Where to learn about parabolic Hölder spaces and when to use them

Is there a good resource from where I can learn about parabolic Hölder spaces? I see quite a few different definitions of this space in different papers. I am clueless about why, for example, one may want your initial condition (or solution) to some PDE to be in the space $C^{k+2, \alpha}$. This is more restrictive than the space $C^{k+2}$ and I don't know what the payoff is.

How do I know which function spaces to use?

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Considering how much of PDE theory concerns finding suitable function spaces to use for a given equation, I doubt that there's any easy answer to "how do I know which function spaces to use" besides (a) experience (someone else used it and it worked) and (b) extreme cleverness (sadly, nothing that I am qualified to explain), plus a small dose of scaling considerations. – Willie Wong Jun 26 '12 at 15:38
One simple reason why you get a payoff is the following "gain of derivatives" (let's consider the elliptic case, the parabolic case is similar): if $\Delta u=f$ in some domain and $f\in C^k$ it does not follow that $u\in C^{k+2}_{loc}$. However, if $f\in C^{k,\alpha}$ and $0<\alpha<1$, then it does follow that $u\in C^{k+2,\alpha}_{loc}$ by Schauder estimates. – YangMills Jun 26 '12 at 16:10
A widely used reference for parabolic PDE is Friedman's "PDE of Parabolic Type" which might be of some help. ams.org/mathscinet-getitem?mr=181836 – Otis Chodosh Jun 26 '12 at 16:41
Thanks for the responses. I'll look for that book in the library. – user24394 Jun 27 '12 at 8:36
@quentinknight a more recent book on Holder space theory in PDE is N. V. Krylov - Lectures on Elliptic and Parabolic Equations in Hölder Spaces books.google.com/… – Andrew Jul 7 '12 at 16:23

The book by N. Krylov is a great reference, but there are some other online references that may help with understanding:

• Lectures on Elliptic and Parabolic Equations in Hölder spaces by N. Krylov

• Generalized Lagrangian Mean Curvature Flow in Almost Calabi-Yau Manifolds doctoral thesis by T. Behrndt

http://people.maths.ox.ac.uk/joyce/theses/BehrndtDPhil.pdf

• Local Existence, Uniqueness and Smooth Dependence for Nonsmooth Quasilinear Parabolic Problems by J. Griepentrog and L. Recke

• Schauder and $L^p$ Estimates for Parabolic Systems via Campanato Spaces by W. Schlag