Local growth envelopes of spaces of generalized smoothness: the critical case. (English) Zbl 1076.46025

Caetano, António M.; Moura, Susana D.

Math. Inequal. Appl. 7, No. 4, 573-606 (2004).

The concept of local growth envelope of a quasi-normed function space is applied to the spaces of Besov and Triebel-Lizorkin type of generalized smoothness \((s,\Psi)\) in the critical case \(s=n/p\), where \(s\) stands for the main smoothness, \(\Psi\) is a perturbation and \(p\) stands for integrability. The expression obtained for the behaviour of the local growth envelope functions shows the possibility to be generalized to a form unifying both the critical \((s=n/p)\) and the subcritical \((s<n/p)\) case.

Reviewer: Meng Wang (Hangzhou)

MSC:

46E35

Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

Besov space; Triebel-Lizorkin space; critical case; local growth envelope