An Inequality in Weighted Campanato Spaces with Applications

Abstract

For 0 < β <1, 0 < p < 1, and ω ∈ A1(ℝn), a version of the John–Nirenberg inequality suitable for the weighted Campanato spaces L(β, p,ω) is established. Further, we show that L(β, p,ω) are independent of the scale p ∈ (0,∞) in the sense of norm when 0 < β < 1 and ω ∈ A1(ℝn). As an application we characterize these spaces by the boundedness of the commutators [b, Iα]i (i = 1, 2), generated by bilinear fractional integral operators of Adams type Iα and the symbol b, from Lp1 (ω) × Lp2 (ω) to Lq(ω1−(1−α/n)q) for 0 < α < 2n, p1, p2 ∈ (1,∞), q ∈ (0,∞), 1/q = 1/q1 + 1/q2 − (α + β)/n and ω ∈ A1(ℝn).