Continuity envelopes for the spaces of generalised smoothness B (s,Ψ) pq (R n) and F (s,Ψ) pq (R n) are studied in the so-called supercritical case s = 1 + n/p, paralleling recent developments for a corresponding limiting case for local growth envelopes of spaces of such a type. In addition, the power of the concept is used in proving conditions for some… (More)
We prove sharp embeddings of Besov spaces B 0,b p,r involving only a slowly varying smoothness b into Lorentz-Karamata spaces. As consequences of our results, we obtain the growth envelope of the Besov space B 0,b p,r .
The concept of local growth envelope (E LG A, u) of the quasi-normed function space A is applied to the Besov spaces of generalized smoothness B σ,N p,q (R n).
We characterize local embeddings of Besov spaces B 0,b p,r involving only a slowly varying smoothness b into classical Lorentz spaces. These results are applied to establish sharp local embeddings of Besov spaces in question into Lorentz-Karamata spaces. As consequence of these results, we are able to determine growth envelopes of spaces B 0,b p,r and to… (More)
We use Kolyada's inequality and its converse form to prove sharp embeddings of Besov spaces B 0,β p,r (involving the zero classical smoothness and a logarithmic smoothness with the exponent β) into Lorentz-Zygmund spaces. We also determine growth envelopes of spaces B 0,β p,r. In distinction to the case when the classical smoothness is positive, we show… (More)