Analysis of some injection bounds for Sobolev spaces by wavelet decomposition

We consider the Sobolev spaces $H^s(\Omega )$ and $H_0^s(\Omega )$ and the Besov spaces $B_{2,\infty }^{1/2}(\Omega )$, where Ω is a sufficiently regular (see Lemma 2) subdomain of ${\mathbb{R}}^2$. It is well known that for the values of $s\in 0,1/2)$ the two Sobolev spaces coincide, with equivalence of the norms, and that the inclusion $B_{2,\infty }^{1/2}(\Omega )?H^s(\Omega )$ holds. The Note is concerned with the explicit analysis of the constants appearing in the continuity bounds for the injections $H^s(\Omega )?H_0^s(\Omega )$ and $B_{2,\infty }^{1/2}(\Omega )?H^s(\Omega )$ and of their dependence on the regularity s of the spaces. The analysis is carried out by using the wavelet characterization of the corresponding norms.