My question: Let $K \subset \mathbb{R}^d$ be compact and $0<\beta<\alpha \leq 1$. Let $\gamma=\theta \alpha+(1-\theta) \beta$ for $\theta \in[0,1]$. Show $$ [f]_{C^{0, \gamma}(K)} \leq[f]_{C^{0, \alpha}(K)}^\theta[f]_{C^{0, \beta}(K)}^{1-\theta} . $$
My solution: Let's first prove the inequality involving the Hölder seminorms. Given $K \subset \mathbb{R}^d$ as a compact set and $0<\beta<\alpha \leq 1$, we define $\gamma=\theta \alpha+(1-\theta) \beta$ for $\theta \in[0,1]$. Our goal is to show that for any function $f$ in these spaces, the inequality $[f]_{C^{0, \gamma}(K)} \leq$ $[f]_{C^{0, \alpha}(K)}^\theta[f]_{C^{0, \beta}(K)}^{1-\theta}$ holds.
Consider two points $x, y \in K$ with $x \neq y$. By the definition of Hölder seminorms, we analyze the expression $\frac{|f(x)-f(y)|}{|x-y|^\gamma}$. Substituting the value of $\gamma$, we get $\frac{|f(x)-f(y)|}{|x-y|^{\theta \alpha+(1-\theta) \beta}}$. This can be rewritten as $\frac{|f(x)-f(y)|}{|x-y|^{\alpha \theta}|x-y|^{\beta(1-\theta)}}$.
Applying Hölder's inequality to this expression, we have $\frac{|f(x)-f(y)|}{|x-y|^{\alpha \theta}|x-y|^{\beta(1-\theta)}} \leq$ $\left(\frac{|f(x)-f(y)|}{|x-y|^\alpha}\right)^\theta\left(\frac{|f(x)-f(y)|}{|x-y|^\beta}\right)^{1-\theta}$. Taking the supremum over all such $x, y$ in $K$, we establish the desired inequality: $[f]_{C^{0, \gamma}(K)} \leq[f]_{C^{0, \alpha}(K)}^\theta[f]_{C^{0, \beta}(K)}^{1-\theta}$.
Next, we show that the unit ball of $C^{0, \alpha}(K)$ is compact in $C^{0, \beta}(K)$. To do this, we use the Arzelá-Ascoli theorem. The theorem states that a set of functions is compact if it is uniformly bounded and equicontinuous. Functions in the unit ball of $C^{0, \alpha}(K)$ are uniformly bounded by definition. They are also equicontinuous since for any $f$ in this ball, $|f(x)-f(y)| \leq|x-y|^\alpha$ for all $x, y \in K$. Thus, these functions satisfy the conditions of the Arzelá-Ascoli theorem, and hence, the unit ball of $C^{0, \alpha}(K)$ is precompact in the uniform topology. Since these functions also have a uniform Hölder constant, the set is compact in $C^{0, \beta}(K)$ for $\beta<\alpha$.
-- I would love any advice or if you could spot any mistakes! :D