Sharp embeddings of Besov-type spaces

We prove sharp embeddings of Besov spaces Bp,rσ,b(R) involving both the classical smoothness σσ and a slowly varying smoothness b   into Lorentz–Karamata spaces. Our methods are quite elementary, we use neither the interpolation theory nor the atomic decomposition of spaces in question. We cover both sub-limiting and limiting cases and we determine growth envelopes of spaces Bp,rσ,b(R).