ABSTRACT
We study continuity envelopes, introduced in Chapters 5 and 6, in the context of function spaces of type Asp,q, where 0 < p ≤ ∞ (with p <∞ in the F -case), 0 < q ≤ ∞, and np ≤ s ≤ np + 1.
We deal with the super-critical case of spaces of type Asp,q as introduced in Figure 11, i.e., np < s ≤ np+1. In view of Proposition 7.14 and (7.47) such spaces can be embedded into C. Hence it is reasonable to study their continuity envelope function. On the other hand, when s > np+1, we may conclude that Asp,q are continuously embedded in Lip1, see (7.45) and (7.46), so that by Proposition 5.3(ii) the corresponding continuity envelope functions are bounded and thus of no further interest. We postpone the borderline case s = np + 1 to the next section.
