If $L^{\infty}$-norm of a set of Holder continuous functions is bounded, then does it imply that its $L^p$-norm is also bounded?

Question

I have a set of real valued functions defined on a bounded open subset $\Omega \subset \mathbb{R}^m$, such that each function $f$ is Holder continuous of modulus $\gamma\ge1-\frac{m}{p},p>m$. It is known that $$\|f\|_{L^{\infty}(\Omega)} \le \delta\forall f\in S$$ Can we say that there exists a $\epsilon \in \mathbb{R}^+$ such that $$\|f\|_{L^p(\Omega)}\le \epsilon \forall f\in S$$

Answer 1

$\|f\|_p \leq \|f\|_{\infty} \mu (\Omega)^{1/p}$ so we can take $\epsilon =\delta \mu (\Omega)^{1/p}$.

I have used $\mu$ to denote the Lebesgue measure on $\mathbb R^{n}$.