Besov spaces with positive smoothness on RnRn, embeddings and growth envelopes

We characterise Besov spaces with positive smoothness on RnRn, obtained by different approaches. First we present two settings Bp,qs(Rn), Bp,qs(Rn) associated to definitions by differences and Fourier-analytical methods and give an equivalent characterisation in terms of subatomic decompositions for the spaces Bp,qs. We study their connections and diversity, as well as embeddings between Besov spaces and into Lorentz spaces. Secondly, we determine their growth envelopes EG(Bp,qs(Rn)) for 00s>0, and finally discuss some applications.