Completeness of Muckenhoupt classes
The authors show that the Hausdorff distance between compact sets and the Kantorovich distance between measures provide an adequate setting to describe the convergence in the class of Muckenhoupt weights. They first recall that the Cantor set and its standard approximation can be described as a convergence result of \((Y_n,\mu_n)\to (C,\mu)\) where one considers the distance \(d_{HK}((Y_n,\mu_n), (C,\mu))\) given by the sum of the Hausdorff distance \(d_H(Y_n,C)\) and the Kantorovich-Rubinshtein distance \(d_k(\mu_n,\mu)\). Then they study in this particular case the weighted situation \(w_n(y)=|y|^\alpha, y\in Y_n\) showing that, for \(s=\frac{\log 2}{\log 3}\) and \(-s<\alpha<s(p_1)\), the terms \(w_n\) become \(A_p\)-weights with a uniform constant. The aim of the note is actually to show that in the general setting of doubling metric compact spaces \((X, d)\), given a sequence of compact probabilistic subspaces \((Y_n,\mu_n)\) satisfying the doubling condition \(0<\mu_n(B_d(y,2r))\leq A\mu_n(B_d(y,r))\) for all \(y\in Y_n, r>0\) and converging to \((Y,\mu)\) in the above distance \(d_{HK}\) and given a sequence of weights \(w_n\in A_p(Y_n,\mu_n)\) with uniform Muckhenhoupt constant \(C\), normalized to have integral \(1\), and weak*-converging to a measure \(\nu\), one has that the limit measure becomes an \(A_p\)-weight, that is \(d\nu=wd\mu\) with \(w\in A_p(Y,\mu)\).
scientific article; zbMATH DE number 5629195
scientific article