Sharp embeddings of Besov-type spaces

We prove sharp embeddings of Besov spaces B p , r σ , b ( R ) involving both the classical smoothness σ 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 B p , r σ , b ( R ) .