Nonhomogeneous $T(1)$ Theorem on Product Quasimetric Spaces

TL;DR

This paper establishes a non-homogeneous $T(1)$ theorem for product quasimetric spaces with measures that are not necessarily doubling but have controlled quasiball sizes, advancing harmonic analysis in complex metric settings.

Contribution

It extends the $T(1)$ theorem to non-homogeneous product quasimetric spaces with measures satisfying upper size control, broadening applicability.

Findings

Proves a non-homogeneous $T(1)$ theorem in product quasimetric spaces.

Handles measures that are not doubling but have controlled quasiball sizes.

Provides tools for analysis in complex metric measure spaces.

Abstract

In this paper, we provide a non-homogeneous $T(1)$ theorem on product spaces $(X_1 \times X_2, \rho_1 \times \rho_2, \mu_1 \times \mu_2)$ equipped with a quasimetric $\rho_1 \times \rho_2$ and a Borel measure $\mu_1 \times \mu_2$, which, need not be doubling but satisfies an upper control on the size of quasiballs.