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.

中文翻译：

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平均积分算子的加权估计

让1 < p ≤ q <+∞，让V，W是在（0，+∞）满足权重：（*）v（X）X ^ρ相当于一个非递减函数上（0，+∞）为对于所有x \ in（0，+ \ infty），大约ρ≥0且\（[w（x）x] ^ {1 / q} \大约[v（x）x] ^ {1 / p}。我们证明，如果平均算子\（（AF）（X）：= \压裂{1} {X} \ INT_0 ^ XF（T）dt的\） ，X ∈（0，+∞）时，从加权界将Lebesgue空间L ^p（（0，+∞）; v）放入加权Lebesgue空间L ^q（（0，+∞），w）中，则存在ε_p ∈（0，p - 1），使得操作者甲还从空间界定大号 ^P-ε（（0，+∞）; V（x）的 ^1个+δ X ^γ到空间大号 ^{q - εq/ P}（（0，+∞）; W（x）的 ^{1 +δ} X ^{δ（1- q / p）} X ^{γ q / p}）的所有ε，δ，γ∈[0，ε ₀）。相反，假设运算符\（A：L ^ {p-\ varepsilon}（（0，+ \ infty）; v（x）^ {1 + \ delta} x ^ \ gamma）\至L ^ {q- \ varepsilon q / p}（（0，+ \ infty）; w（x）^ {1 + \ delta} x ^ {\ delta（1-q / p）} x ^ {\ gamma q / p}）\ ）有界ε∈[0，p-1），δ≥0和γ≥0，我们证明算子A也从空间L ^p（（0，+∞）; v）进入空间L ^q（（0，+∞）; w）。尤其是，我们的结果表明，拥有（*）且操作符A约束在空间L ^p（（0，+∞）; v）上的权重v具有与A ^p -class相似的性质B. Muckenhoupt。