Some new characterizations of the class of positive measures γ on$R$^{$n$}such that$H$_{$p$}^{$l$}∉L_{$p$}(γ) are given where$H$_{$p$}^{$l$}(1<$p$<∞ 0<$l$∞) is the space of Bessel potentials This imbed ding as well as the corresponding trace inequality $$||J_l u||_{L_p (\gamma )} \leqslant C||u||_{L_p } $$ for Bessel potentials$J$_{$l$}=(1-Δ)^{-1/2}is shown to be equivalent to one of the following conditions(a)$J$_{$l$}($J$_{$l$γ})^{$p$}≤$CJ$_{$lγ$}a e(b)$M$_{$l$}($M$_{$l$γ})^{$p’$}≤$CM$_{$lγ$}a e(c)For all compact subsets$E$of$R$^{$n$} $$\int_E {(J_{l\gamma } )^p dx} \leqslant C{\text{ }}cap (E H_p^l )$$ where 1/$p$+1/$p'$=1$M$_{$l$}is the fractional maximal operator and cap ($H$_{$p$}^{$l$}) is the Bessel capacity In particular it is shown that the trace inequality for a positive measure \gg holds if and only if it holds for the measure$(J$_{$l\gg$})^{$p'$}$dx$Similar results are proved for the Riesz potentials$I$_{$l$γ}=|$x$|^{$l-n$}* γ

This research provides insights for the separation of cryptocurrencies from other assets. Using dimensionality reduction techniques, we show that most of the variation among cryptocurrencies, stocks, exchange rates, commodities, bonds, and real estate indexes can be explained by the tail, memory and moment factors of their log-returns. By applying various classification methods, cryptocurrencies are categorized as a separate asset class, mainly due to the tail factor. The main result is the complete separation of cryptocurrencies from the other asset types, using the Maximum Variance Components Split method. Additionally, we show that cryptocurrencies tend to exhibit similar characteristics over time and become more distinguished from other asset classes (synchronic evolution).

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