A Note on Campanato Spaces and Their Applications

In this paper, we obtain a version of the John–Nirenberg inequality suitable for Campanato spaces C_p,β with 0 < p < 1 and show that the spaces C_p,β are independent of the scale p ∈ (0,∞) in sense of norm when 0 < β < 1. As an application, we characterize these spaces by the boundedness of the commutators b,B _α ] _j (j = 1, 2) generated by bilinear fractional integral operators B _α and the symbol b acting from L_p1 × L_p2 to L ^q for p1, p2 ∈ (1,∞), q ∈ (0,∞) and 1/q = 1/p1 + 1/p2 ? (α + β)/n.