For the maximal operator $M$ on $\mathbb{R}^{d}$, and $ 1 < p,\rho < \infty$, there is a finite constant $D=D _{p, \rho }$ so that this holds. For all weights $w,\sigma$ on $\mathbb{R}^{d}$, the operator $M(\sigma \cdot)$ is bounded from $L^{p}(\sigma )\to L^{p}(w)$ if and only if the pair of weights $(w,\sigma)$ satisfy the two weight $A_{p}$ condition, and this testing inequality holds: $$\int_{Q} M(\sigma \mathbf 1_{Q})^{p} dw \lesssim \sigma(Q),$$ for all cubes $Q$ for which there is a cube $P\supset Q$ satisfying $\sigma(P) \leq D\sigma(Q)$, and $\ell(P) \geq \rho \ell(Q)$. This was recently proved by Kangwei Li and Eric Sawyer. We give a short proof, which is easily seen to hold for several closely related operators.

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正运算符的两个权重不等式：立方体加倍

对于$ \ mathbb {R} ^ {d} $上的最大运算符$ M $，以及$ 1 <p，\ rho <\ infty $，存在一个有限常数$ D = D _ {p，\ rho} $这样就成立了。对于$ \ mathbb {R} ^ {d} $上的所有权重$ w，\ sigma $，运算符$ M（\ sigma \ cdot）$的范围从$ L ^ {p}（\ sigma）\到L ^ {p}（w）$当且仅当一对权重$（w，\ sigma）$满足两个权重$ A_ {p} $的条件，并且该测试不等式成立时：$$ \ int_ {Q} M（ \ sigma \ mathbf 1_ {Q}）^ {p} dw \ lesssim \ sigma（Q），$$表示所有多维数据集$ Q $，其中有一个多维数据集$ P \ setup Q $满足$ \ sigma（P）\ leq D \ sigma（Q）$和$ \ ell（P）\ geq \ rho \ ell（Q）$。李康伟和埃里克·索耶（Eric Sawyer）最近证明了这一点。我们提供了一个简短的证明，对于几个紧密相关的运营商来说，很容易找到它。