In this paper we construct a wavelet basis in \(L^{2}({\mathbb {R}}^{n};\mu )\) possessing vanishing moments of a fixed order for a general locally finite positive Borel measure \(\mu \). The approach is based on a clever construction of Alpert in the case of the Lebesgue measure that is appropriately modified to handle the general measures considered here. We then use this new wavelet basis to study a two-weight inequality for a general Calderón–Zygmund operator on \({\mathbb {R}}\) and conjecture that under suitable natural conditions, including a weaker energy condition, the operator is bounded from \(L^{2}({\mathbb {R}};\sigma )\) to \(L^{2}({\mathbb {R}};\omega )\) if certain stronger testing conditions hold on polynomials. An example is provided showing that this conjecture is logically different from existing results in the literature.

在本文中，我们在\（L ^ {2}（{\ mathbb {R}} ^ {n}; \ mu）\）中构造一个小波基，它具有一般局部有限正Borel测度\的固定阶数的消失矩。（\ mu \）。该方法基于Lebesgue措施的Alpert的巧妙构造，该措施经过适当修改以处理此处考虑的一般措施。然后，我们使用这个新的小波基础研究\（{\ mathbb {R}} \）上的一般Calderón–Zygmund算子的二重不等式，并推测在适当的自然条件（包括较弱的能量条件）下，该算子为从有界\（L ^ {2}（{\ mathbb {R}}; \西格玛）\）到\（L ^ {2}（{\ mathbb {R}}; \欧米加）\）如果某些多项式具有更强的测试条件。提供了一个示例，表明该推测在逻辑上与文献中的现有结果不同。