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On fractional Laplacians

We compare two natural types of fractional Laplacians $(-Δ)^s$, &#34;Navier&#34; and &#34;Dirichlet&#34; ones. We show that for $0<s<1$ their difference is positive definite and positivity preserving. Then we prove the coincidence of the Sobolev constants for these two fractional Laplacians.