Let σ and ω be locally finite Borel measures on ℝ^d, and let \(p\in (1,\infty )\) and \(q\in (0,\infty )\). We study the two-weight norm inequality \( \lVert T(f\sigma ) \rVert _{L^{q}(\omega )}\leq C \lVert f \rVert _{L^{p}(\sigma )}, \quad \text {for all} f \in L^{p}(\sigma ), \) for both the positive summation operators T = T_λ(⋅σ) and positive maximal operators T = M_λ(⋅σ). Here, for a family {λ_Q} of non-negative reals indexed by the dyadic cubes Q, these operators are defined by \( T_{\lambda }(f\sigma ):={\sum }_{Q} \lambda _{Q} \langle {f}\rangle ^{\sigma }_{Q} 1_{Q} \quad \text { and } \quad M_{\lambda }(f\sigma ):=\sup _{Q} \lambda _{Q} \langle {f}\rangle ^{\sigma }_{Q} 1_{Q}, \) where \(\langle {f}\rangle ^{\sigma }_{Q}:=\frac {1}{\sigma (Q)} {\int \limits }_{Q} |f| d \sigma \). We obtain new characterizations of the two-weight norm inequalities in the following cases: (1) For T = T_λ(⋅σ) in the subrange q < p. Under the additional assumption that σ satisfies the \(A_{\infty }\) condition with respect to ω, we characterize the inequality in terms of a simple integral condition. The proof is based on characterizing the multipliers between certain classes of Carleson measures. (2) For T = M_λ(⋅σ) in the subrange q < p. We introduce a scale of simple conditions that depends on an integrability parameter and show that, on this scale, the sufficiency and necessity are separated only by an arbitrarily small integrability gap. (3) For the summation operators T = T_λ(⋅σ) in the subrange 1 < q < p. We characterize the inequality for summation operators by means of related inequalities for maximal operators T = M_λ(⋅σ). This maximal-type characterization is an alternative to the known potential-type characterization. The subrange of the exponents q < p appeared recently in applications to nonlinear elliptic PDE with \(\lambda _{Q} = \sigma (Q) |Q|^{\frac {\alpha }{d}-1}\), α ∈ (0, d). In this important special case T_λ is a discrete analogue of the Riesz potential \(I_{\alpha }=(-{\Delta })^{-\frac {\alpha }{2}}\), and M_λ is the dyadic fractional maximal operator.

让σ和ω是对ℝ局部有限波雷尔措施^d，让\（P \在（1，\ infty）\）和\（在（0 q \，\ infty）\） 。我们研究二重范数不等式\（\ lVert T（f \ sigma）\ rVert _ {L ^ {q}（\ omega}} \ leq C \ lVert f \ rVert _ {L ^ {p}（\ sigma ）}，\四\文本{对于所有} F到\在L ^ {p}（\西格玛），\）用于正求和操作符都Ť = Ť _λ（⋅ σ）和阳性极大算Ť =中号_λ（⋅ σ）。在这里，为家庭{ λ _Q由二元立方体索引的非负实数} Q，这些运算符由\（T _ {\ lambda}（f \ sigma）：= {\ sum __ {Q} \ lambda _ {Q} \ langle {f} \ rangle ^ {\ sigma __ {Q} 1_ {Q} \ quad \ text {和} \ quad M _ {\ lambda}（f \ sigma）：= \ sup _ {Q} \ lambda _ {Q} \ langle {f} \ rangle ^ {\ sigma} _ {Q} 1_ {Q}，\）其中\（\ langle {f} \ rangle ^ {\ sigma} _ {Q}：= \ frac {1} {\ sigma（Q）} {\ int \ limits} _ {Q} | f | d \ sigma \）。我们在下列情况下获得的两个重量范数不等式的新表征：（1）对于Ť = Ť _λ（＆CenterDot;＆σ）在子范围q < p。在σ满足ω的\（A _ {\ infty} \）条件的附加假设下，我们用一个简单的积分条件来描述不等式。证明基于表征某些类别的Carleson测度之间的乘数。（2）对于Ť =中号_λ（＆CenterDot;＆σ）在子范围q < p。我们介绍了一个取决于可积性参数的简单条件量表，并表明在此量表上，充裕度和必要性仅由任意小的可积度差距分开。（3）为求和操作符Ť = Ť _λ（＆CenterDot;＆σ）在子范围1 < q < p。我们通过对最大相关运营商不等式手段表征为求和操作符不等式Ť =中号_λ（＆CenterDot;＆σ）。这种最大类型的表征是已知电势类型表征的替代方案。q < p的子范围最近出现在具有\（\ lambda _ {Q} = \ sigma（Q）| Q | ^ {\ frac {\ alpha} {d} -1} \）的非线性椭圆PDE的应用中，α＆Element;（0，d）。在此重要的特殊情况Ť _λ是Riesz位电位的离散的类似物\（I _ {\阿尔法} =（ - {\德尔塔}）^ { - \压裂{\阿尔法} {2}} \） ，和中号_λ 是二进式分数最大运算符。