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 εp ∈ (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.
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The paper was supported by the grants nos. 201/05/2033 and 201/08/0383 of the Czech Science Foundation and by the Institutional Research Plan no. AV0 Z10190503 of the Academy of Sciences of the Czech Republic
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Opic, B., Rákosník, J. Weighted estimates for the averaging integral operator. Collect. Math. 61, 253–262 (2010). https://doi.org/10.1007/BF03191231
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DOI: https://doi.org/10.1007/BF03191231