The Averaging Integral Operator between Weighted Lebesgue Spaces and Reverse Hölder Inequalities

Let 1 < p ≤ q < +∞ and let v, w be weights on (0,+∞) satisfying: v(x)xρ is equivalent to a non-decreasing function on (0,+∞) for some ρ ≥ 0; [w(x)x] ≈ [v(x)x] for all x ∈ (0,+∞). Let A be the averaging operator given by (Af)(x) := 1 x R x 0 f(t) dt, x ∈ (0,+∞). First, we prove that the operator A : L((0,+∞); v)→ L((0,+∞); v) is bounded if and only if the operator A : L((0,+∞); v)→ L((0,+∞);w) is bounded. Second, we show that the boundedness of the averaging operator A on the space Lp((0,+∞); v) implies that, for all r > 0, the weight v1−p satisfies the reverse Hölder inequality over the interval (0, r) with respect to the measure dt, while the weight v satisfies the reverse Hölder inequality over the interval (r,+∞) with respect to the measure t−p dt. As a corollary, we obtain that the boundedness of the averaging operator A on the space Lp((0,+∞); v) is equivalent to the boundedness of the averaging operator A on the space Lp((0,+∞); v1+δ) for some δ > 0.