For real numbers $\alpha > \beta$, we know there is a continuous embedding of Besov spaces $B^\alpha_{\infty,\infty}\subset B^\beta_{\infty,\infty}$. We take the closure of the intersection $C^{\infty} \cap B^\beta_{\infty,\infty}$ in $B^\beta_{\infty,\infty}$, where $C^{\infty}$ is the space of smooth functions, and then intersect this closure with $B^\alpha_{\infty,\infty}$, how can we show that the result is dense in $B^\alpha_{\infty,\infty}$?
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$\begingroup$ Are the Besov spaces you consider defined on $\mathbb{R}^d$? If not, what kind of domains do you consider? $\endgroup$– Onur OktayCommented Dec 21, 2021 at 21:59
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$\begingroup$ @OnurOktay Yes, the domain is $\mathbb{R}^d$ $\endgroup$– InuyashaCommented Dec 22, 2021 at 14:08
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$\begingroup$ $B^s_{\infty,\infty}(\mathbb{R}^d)$ are the Hölder-Zygmund spaces. Please see mathoverflow.net/questions/244377/… $\endgroup$– Onur OktayCommented Dec 22, 2021 at 15:46
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$\begingroup$ Please also see Hajlasz's reply in mathoverflow.net/questions/29869/… $\endgroup$– Onur OktayCommented Dec 22, 2021 at 16:01
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$\begingroup$ For the equivalence of $B^s_{\infty,\infty}$ and the Hölder Zygmund spaces $C^s$, please see the monograph by Triebel "Theory of Function Spaces, I", Sections 2.3.5 and 2.5.7 $\endgroup$– Onur OktayCommented Dec 22, 2021 at 16:04
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