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I recall that $f\in \mathcal C^\alpha ([0,1[)$ where $\alpha \in (0,1)$ if $$[f]_\alpha :=\sup_{x,y\in\mathbb [0,1[}\frac{|f(x)-f(y)|}{|x-y|^\alpha }<\infty .$$

Does those space are dense in $L^p$ space ? Or in Schwarz space ? I'm asking this question because they use this result at the beginning of the proof page 3 here.

If yes, can we generalize this to $\mathbb R^n$ ? i.e. Is $\mathcal C^\alpha (\mathbb R^n)$ dense in $L^p(\mathbb R^n)$ ?

MSE
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    I think the answer to your question follows from the facts that the compactly supported smooth functions are dense in $L^p$ and that smooth functions are Hölder continuous. – Christian Jul 05 '16 at 11:08

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