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Background: The proof of Theorem 6.4 in http://mate.dm.uba.ar/~jrossi/Fractional-1-lapla-07_02_2015.pdf, I want to use the density that $C^\infty(\Omega) \cap W^{r_0,q_0}(\Omega) \cap L^2(\Omega)$ is dense in $W^{s,1}(\Omega) \cap L^2(\Omega)$, where $1<q_0<\frac{N}{N+s-1}$, $r_0 = N+s-\frac{N}{q_0}<1$ to get the conclusion like (3.11).

Let $\Omega$ be a nonempty bounded open set in $\mathbb{R}^{n}$, $1 < p_1 < + \infty$, $1 \leq p_2 < + \infty$, $0<s_1 \leq 1$, $0<s_2 \leq 1$.
Q1: If the density is true, how to prove it? If not, what condition is necessary to make it true?
Q2: Let $p_2<2$, is $C^\infty(\Omega) \cap W^{s_1,p_1}(\Omega) \cap L^2(\Omega)$ dense in $W^{s_2,p_2}(\Omega) \cap L^2(\Omega)$ if $W^{s_1,p_1}(\Omega) \subset W^{s_2,p_2}(\Omega)$? If it is true, how to prove it?

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  • $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Community Bot
    Commented Feb 5, 2023 at 10:15
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    $\begingroup$ This post was asked again by the OP at math.stackexchange.com/q/4632827/18398 $\endgroup$
    – JRN
    Commented Feb 6, 2023 at 3:50

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