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

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*Krylov, N. V., Lectures on elliptic and parabolic equations in Sobolev spaces, Graduate Studies in Mathematics 96. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4684-1/hbk). xviii, 357 p. (2008). ZBL1147.35001.


*Behrndt, T.  Generalized Lagrangian Mean Curvature Flow in Almost Calabi-Yau Manifolds, doctoral thesis. PDF link.


*Griepentrog, Jens André; Recke, Lutz, Local existence, uniqueness and smooth dependence for nonsmooth quasilinear parabolic problems, J. Evol. Equ. 10, No. 2, 341-375 (2010). ZBL1239.35084.


*Ibrahim, Hassan, A generalization of a logarithmic Sobolev inequality to the Hölder class, J. Funct. Spaces Appl. 2012, Article ID 148706, 15 p. (2012). ZBL1242.46041.


*Schlag, Wilhelm, Schauder and $L^ p$ estimates for parabolic systems via Campanato spaces, Commun. Partial Differ. Equations 21, No. 7-8, 1141-1175 (1996). ZBL0864.35023. Link on author's website here.
