Weighted estimates for the averaging integral operator

Abstract

Let 1 <p≤q<+∞ and letv, w be weights on (0, + ∞) satisfying: (*)v(x)x ^ρ is equivalent to a non-decreasing function on (0, +∞) for someρ ≥ 0 and$[w(x)x]^{1/q} \approx [v(x)x]^{1/p} for all x \in (0, + \infty ).$ We prove that if the averaging operator$(Af)(x): = \frac{1}{x}\int_0^x f (t) dt$,x ∈ (0, + ∞), is bounded from the weighted Lebesgue spaceL ^p ((0, + ∞);v) into the weighted Lebesgue spaceL ^q((0, + ∞),w), then there exists ε_p ∈ (0,p − 1) such that the operatorA is also bounded from the spaceL ^p-ε ((0, + ∞);v(x) ^1+δ x ^γ into the spaceL ^q-εq/p((0, + ∞);w(x) ^1+δ x ^δ(1-q/p) x ^γq/p) for all ε, δ, γ ∈ [0, ε₀). Conversely, assuming that the operator$A : L^{p - \varepsilon } ((0, + \infty ); v(x)^{1 + \delta } x^\gamma ) \to L^{q - \varepsilon q/p} ((0, + \infty ); w(x)^{1 + \delta } x^{\delta (1 - q/p)} x^{\gamma q/p} )$ is bounded for some ε ∈ [0,p−1), δ ≥ 0 and γ ≥ 0, we prove that the operatorA is also bounded from the spaceL ^p((0, + ∞);v) into the spaceL ^q((0, + ∞);w). In particular, our results imply that the class of weightsv for which (*) holds and the operatorA is bounded on the spaceL ^p((0, + ∞);v) possesses properties similar to those of theA ^p-class of B. Muckenhoupt.