Morrey smoothness spaces:A new approach

In the recent years,the so-called Morrey smoothness spaces attracted a lot of interest.They can(also)be understood as generalisations of the classical spaces A_(p,q)^(s)(R^(n))with A∈{B,F}in R^(n),where the parameters satisfy s∈R(smoothness),0<p∞(integrability)and 0<q∞(summability).In the case of Morrey smoothness spaces,additional parameters are involved.In our opinion,among the various approaches at least two scales enjoy special attention,also in view of applications:the scales A_(p,q)^(s)(R^(n))with A∈{N,E}and u≥p,and A_(p,q)^(s),τ(R^(n))with A∈{B,F}andτ≥0.We reorganise these two prominent types of Morrey smoothness spaces by adding to(s,p,q)the so-called slope parameter e,preferably(but not exclusively)with-n e<0.It comes out that|e|replaces n,and min(|e|,1)replaces 1 in slopes of(broken)lines in the(1/p,s)-diagram characterising distinguished properties of the spaces A_(p,q)^(s)(R^(n))and their Morrey counterparts.Special attention will be paid to low-slope spaces with-1<e<0,where the corresponding properties are quite often independent of n∈N.Our aim is two-fold.On the one hand,we reformulate some assertions already available in the literature(many of which are quite recent).On the other hand,we establish on this basis new properties,a few of which become visible only in the context of the offered new approach,governed,now,by the four parameters(s,p,q,e).

Some Specific Unboundedness Property in Smoothness Morrey Spaces. The Non-existence of Growth Envelopes in the Subcritical Case

Abstract We study smoothness spaces of Morrey type on Rn and characterise in detail those situa s,r n s n tions when such spaces of type Ap,q^s,r（Rn ） or A u^sp,q（R ） 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 Mu,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.