Some specific unboundedness property in smoothness Morrey spaces. The non-existence of growth envelopes in the subcritical case

We study smoothness spaces of Morrey type on R ⁿ and characterise in detail those situations when such spaces of type A _p,q ^s,τ (R ⁿ ) or A _u,p,q ^s (R ⁿ ) are not embedded into L _∞(R ⁿ ). We can show that in the so-called sub-critical, proper Morrey case their growth envelope function is always infinite which is a much stronger assertion. The same applies for the Morrey spaces M _u,p (R ⁿ ) with p < u. This is the first result in this direction and essentially contributes to a better understanding of the structure of the above spaces.