A Note on Campanato Spaces and Their Applications

In this paper, we obtain a version of the John–Nirenberg inequality suitable for Campanato spaces $\mathcal{C}_{p,\beta}$ with $0<p<1$ and show that the spaces $\mathcal{C}_{p,\beta}$ are independent of the scale $p\in (0,\infty)$ in sense of norm when $0<\beta<1$. As an application, we characterize these spaces by the boundedness of the commutators $[b,B_{\alpha}]_{j}$ $(j=1,2)$ generated by bilinear fractional integral operators $B_{\alpha}$ and the symbol $b$ acting from $L^{p_{1}}\times L^{p_{2}}$ to $L^{q}$ for $p_{1},p_{2}\in(1,\infty), q\in (0,\infty)$ and $1/q=1/p_{1}+1/p_{2}-(\alpha+ \beta)/n$.