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$\newcommand{\RR}{\mathbb R}\newcommand{\NN}{\mathbb N}$

Let $\alpha \in (0, 1)$ and $j \in \NN$. We denote by $H^{j + \alpha} := H^{j + \alpha} ({\RR}^d)$ the space of real-valued functions $f$ on $\RR^d$ which are continuous and whose partial derivatives $D^k f := \frac{\partial^{k_1}}{\partial x_1^{k_1}} \frac{\partial^{k_2}}{\partial x_2^{k_2}} \cdots \frac{\partial^{k_d}}{\partial x_d^{k_d}}$ are bounded for $|k| \le j$ and that admit a finite norm $$ \| f \|_{H^{j + \alpha}} := \sum_{| k | \le j} \| D^\alpha f\|_\infty + \sum_{|k| = j} \sup_{\substack{x, x' \in {\RR}^d \\ |x-x'| \le 1}} \frac{| D^k f (x) - D^k f (x')|}{|x-x'|^\alpha}, $$ where $k = (k_1, \ldots, k_d) \in \NN^d$ is a multi-index of length $|k | := k_1 + \cdots + k_d$.

Let $p \in H^{0 + \alpha}$ be a probability density function (p.d.f.).

Can we approximate $p$ (w.r.t. $\| \cdot \|_{H^{0 + \alpha}}$) by p.d.f.'s in $H^{1 + \alpha}$?

Any reference is greatly appreciated. Thank you so much for your elaboration!

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    $\begingroup$ Approximate in what norm? Also is there any relevance of these functions being pdfs? I guess that's just something like "a non-negative Lebesgue integrable function that integrates to 1" right? $\endgroup$
    – SBK
    Commented Nov 27, 2023 at 14:33
  • $\begingroup$ @SBK You are right! I meant the approximation in the norm $\| \cdot \|_{H^{0 + \alpha}}$. $\endgroup$
    – Akira
    Commented Nov 27, 2023 at 14:35

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No, take any $p$ such that $p(x)=|x|^\alpha$ in some neighbourhood of the origin.

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