Weighted Estimates for the Averaging Integral Operator

Let 1 < p ≤ q < +∞ and let v, w be weights on (0,+∞) satisfying: (?) v(x)xis equivalent to a non-decreasing function on (0,+∞) for some ρ ≥ 0; [w(x)x] ≈ [v(x)x] for all x ∈ (0,+∞). We prove that if the averaging operator (Af)(x) := 1 x R x 0 f(t) dt, x ∈ (0,+∞), is bounded from the weighted Lebesgue space Lp((0,+∞); v) into the weighted Lebesgue space Lq((0,+∞);w), then there exists ε0 ∈ (0, p− 1) such that the operator A is also bounded from the space Lp−ε((0,+∞); v(x)1+δxγ) into the space Lq−εq/p((0,+∞);w(x)1+δxδ(1−q/p)xγq/p) for all ε, δ, γ ∈ [0, ε0). Conversely, assuming that the operator A : Lp−ε((0,+∞); v(x)x)→ Lq−εq/p((0,+∞);w(x)1+δxδ(1−q/p)xγq/p) is bounded for some ε ∈ [0, p−1), δ ≥ 0 and γ ≥ 0, we prove that the operator A is also bounded from the space Lp((0,+∞); v) into the space Lq((0,+∞);w). In particular, our results imply that the class of weights v for which (?) holds and the operator A is bounded on the space Lp((0,+∞); v) possesses similar properties to those of the Ap-class of B. Muckenhoupt.