Weighted Alpert Wavelets

In this paper we construct a wavelet basis in L2(Rn;μ)$$L^{2}({\mathbb {R}}^{n};\mu )$$ possessing vanishing moments of a fixed order for a general locally finite positive Borel measure μ$$\mu $$. 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 R$${\mathbb {R}}$$ and conjecture that under suitable natural conditions, including a weaker energy condition, the operator is bounded from L2(R;σ)$$L^{2}({\mathbb {R}};\sigma )$$ to L2(R;ω)$$L^{2}({\mathbb {R}};\omega )$$ 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.