Two-weight Hardy inequality on topological measure spaces
Year: 2025
Type: article
Abstract: We consider a Hardy type integral operator \( T \) associated with a family of open subsets \( \Omega(t) \) of an open set \( \Omega \) in a Hausdorff topological space \( X \). In the inequality\[\left( \int_\Omega |Tf(x)|^q u(x)\,d\mu(x) \right)^{1/q} \leq C \left( \int_\Omega |f(x)|^p v(x)\,d\nu(x) \right)^{1/p},\]the measures \( \mu, \nu \) are \( \sigma \)-additive Borel measures; the weights \( u, v \) are positive and finite almost everywhere, \( 1 < p < \infty \), \( 0 < q < \infty \), and \( C > 0 \) is independent of \( f, u, v, \mu, \nu \). We find necessary and sufficient conditions for the boundedness and compactness of the operator \( T \) and obtain two-sided estimates for its approximation numbers. All results are proved using domain partitions, thus providing a roadmap for generalizing many one-dimensional results to a Hausdorff topological space.
Source: Eurasian Mathematical Journal
Authors: K.T. Mynbaev, E.N. Lomakina
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4411158655