Definition of Hölder space

Question

I am wondering what is the definition for Hölder space $C^\gamma$ when $\gamma\in \mathbb{N}$.

Let's take the underlying field $\mathbb{R}^d$. Is it $$ C^\gamma = \{f:\ f\in C^{\gamma-1}\}\cap \{f: \partial^\alpha f\in Lip,\ |\alpha| = \gamma-1\} $$ or common space $$C^\gamma = \{f: \|\partial^\alpha f \|_\infty<\infty, |\alpha|\leq \gamma\}.$$

I guess I would be the first one, otherwise I don't see why we need Zygmund space to fill the gap.

Answer 1

Let's focus on spaces rather than notation. We have:

• The space of functions with continuous derivatives of orders $\le k$. Usually denoted $C^k$.
• The space of functions with continuous derivatives of orders $\le k$, where the $k$th derivative is Hölder continuous of order $\alpha\in (0,1]$. Usually denoted $C^{k,\alpha}$ but sometimes also $C^{k+\alpha}$. The latter notation is a bit dangerous because $C^{1+1}$ is not the same as $C^{2}$.
• The space of functions with bounded weak derivatives of orders up to $k$. Usually denoted $W^{k,\infty}$.

We have $C^{k,1}\subseteq W^{k+1,\infty}$, because Lipschitz functions have bounded weak derivatives of first order. If the spaces are considered on a sufficiently nice domain (without cusps, slits etc.) then $C^{k,1} = W^{k+1,\infty}$. In particular, this is the case for $\mathbb R^d$.

I don't see why we need Zygmund space to fill the gap.

Notation is a wrong place to look for a reason why some object was introduced. A major reason to introduce the Zygmund space is the behavior of the Hilbert transform (and other singular integral operators) on $C^{k,\alpha}$. When $0<\alpha<1$, the space is mapped onto itself. When $\alpha=1$, it isn't: e.g., the transform of a Lipschitz function isn't Lipschitz. But the slightly larger Zygmund space turns out to be preserved by the Hilbert transform. Hence, we can conclude that the transform of a Lipschitz function is a member of the Zygmund class.