Bulk/boundary quotients of Gaussian multiplicative chaos measures I: positive moment bounds

This is the first part in a series of papers devoted to studying the tail profiles of the bulk/boundary quotients of Gaussian multiplicative chaos measures appearing in boundary Liouville conformal field theory, for which we establish preliminary joint moment bounds. These moment bounds will serve as a crucial ingredient in establishing the right tail profile of the bulk Gaussian multiplicative chaos measure in subsequent papers. The main idea is to implement the so-called localization trick at the boundary, and we also record a useful generalization of Kahane’s convexity inequality, which is of independent interest.