New function classes of Morrey–Campanato type and their applications

New John–Nirenberg–Campanato‐type spaces related to both maximal functions and their commutators

“…where the supremum is taken over all cubes Q of R n . Very recently, Wang and Shu 22 further showed that if we replace  Q (b) in (iii) by |b| Q , then the above equivalences still hold true. Indeed, they obtain the equivalence between (ii) and (iv) sup…”

“…However, || • || 𝛼,q,s () is not a norm of 𝛼,q,s () because the triangle inequality does not hold true. (ii) Comparing 𝛼,q,s () in Definition 2.1 (ii) with  p,𝜆 () in Wang and Shu, 22 it is easy to find that…”

New John--Nirenberg--Campanato-Type Spaces Related to Both Maximal Functions and Their Commutators

“…Very recently, Wang and Shu [30] further showed that, if we replace M Q (b) in (iii) by |b| Q , then the above equivalence still holds true. Indeed, they obtain the equivalence between (ii) and…”

“…Proof. Since the case p = ∞ was obtained in [30,Theorem 3.6], it suffices to consider the case p ∈ [1, ∞).…”

“…This implies that (i) ⇐⇒ (ii) ⇐⇒ (iii) ⇐⇒ (iv). Moreover, Wang and Shu [30] also studied this phenomenon in the context of Campanato spaces, and introduced a new function spaces of Morrey-Campanato type in [30]. As applications, they obtained an integral characterization of the non-negative Hölder continuous functions.…”