Resumen de Weighted estimates for the averaging integral operator

Bohumír Opic, Jiri Rakosnik

• 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 some ρ ≥ 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, ε_0). 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.