Hölder norm bounded by $L^p-$norm?

Question

Let $C_0^{\alpha}(\mathbb{R})$, $0<\alpha<1$ denote the space of Hölder-continuous functions on $\mathbb{R}$ with compact support.

Is it true that for any $f\in C_b^{\alpha}(\mathbb{R})$ one has $\| f\|_p \leq C \| f\|_{\alpha}$ for any $p$ maybe strongly depending on $\alpha$?. In other words, are there known embeddings from $C_0^\alpha$ to $L^p$ spaces? If not, are there any counterexamples? I have the feeling its true since $C_0^{\alpha}\hookrightarrow C_0^1 \hookrightarrow L^p$ right? But is there a direct proof of the estimate?

Thank you very much!

Answer 1

This is true if $\mathbb R$ is replaced by a closed interval $[a,b]$. To see that this is not true, take $$ f_n(x)=\left\{\begin{array}{clc} 0 & \text{if} & x\le -2n, \\ \frac{x+2n}{n} & \text{if} & -2n\le x\le -n, \\ 1 & \text{if} & -n\le x \le n, \\ \frac{2n-x}{n} & \text{if} & n\le x\le 2n, \\ 0 & \text{if} & x \ge 2n. \end{array} \right. $$ Then $$ \|f_n\|_p>2n\qquad\text{and}\qquad \|f\|_{0,\alpha}=1. $$