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 Rn and characterise in detail those situations when such spaces of type A s,.p,q (Rn) or A su,p,q(Rn) are not embedded into L .(Rn). 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 (Rn) 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.