On a bridge connecting Lebesgue and Morrey spaces in view of their growth properties

We study generalized Morrey spaces M-phi,M-p(R-d) and Besov-Morrey spaces related to them. These spaces appear quite naturally in connection with the study of PDE or to characterize boundedness of certain integral operators. Here, we study unboundedness properties of functions belonging to those spaces in terms of their growth envelopes. This concept has been studied and applied successfully for a variety of smoothness spaces already. Surprisingly, for the generalized Morrey spaces we arrive at three possible cases only, i.e. boundedness, the L-p-behavior or the proper Morrey behavior. These cases are characterized in terms of the limit of phi(t) and t(-d/p)phi(t) as t -> 0(+) and t -> infinity, respectively. For the generalized Besov-Morrey spaces the limit of t(-d/p)phi(t) as t -> 0(+) also plays a role.