A direct proof of the weighted Pólya–Knopp inequality following Carleson’s method - Kondo, Emu; Moritoh, Shinya; Tanaka, Yumi (Comptes Rendus. Mathématique, 2025)
The aim of the paper is to provide a direct proof of the weighted Pólya–Knopp inequality. This inequality (which is a limiting case of the Ariño–Muckenhoupt inequalities), involving non-increasing functions, was initially established by Sbordone–Wik, who proved its validity under the necessary and sufficient condition that the weight satisfies an appropriate doubling condition. Our main contribution is to use Carleson’s approach to Carleman’s inequality in conjunction with Hardy’s lemma and Sbordone–Wik’s doubling condition, in order to obtain the weighted Pólya–Knopp inequality.