Nontriviality of John–Nirenberg–Campanato Spaces

Abstract

Let $p,q\in [1,\infty )$, $\alpha \in {\mathbb {R}}$, and s be a non-negative integer. This article addresses the nontriviality of the John–Nirenberg–Campanato space $JN_{(p,q,s)_\alpha }({\mathcal {X}})$, where ${\mathcal {X}}$ denotes ${\mathbb {R}}^n$ or any given cube of ${\mathbb {R}}^n$. First, the authors obtain some (non-)trivial ranges of $\{p,q,s,\alpha \}$, which sheds some light on the independence of $JN_{(p,q,s)_\alpha }({\mathcal {X}})$ over the second subindex q. Second, for positive $\alpha$, the authors show that $JN_{(p,q,s)_\alpha }({\mathcal {X}})$ is different from the Campanato space via establishing an embedding from the fractional Sobolev space to $JN_{(p,q,s)_\alpha }({\mathcal {X}})$. Third, for negative $\alpha$, the authors show that the Riesz–Morrey space $RM_{p,q,\alpha }({\mathcal {X}})$ is a proper subspace of $JN_{(p,q,s)_\alpha }({\mathcal {X}})$, which gives a negative answer to the conjecture on the equivalence between John–Nirenberg–Campanato spaces and Riesz–Morrey spaces. Moreover, the nontriviality of local John–Nirenberg–Campanato spaces is also presented.