An HK$$_r$$-Integrable Function Which is P$$_s$$-Integrable for no s

Abstract

Given arbitrary $$r\ge 1$$ r ≥ 1 , we construct an HK$$_r$$ r -integrable function which is not P$$_1$$ 1 -integrable. This is an extension of a recently published construction [Musial, P., Skvortsov, V., Tulone, F.: The HK$$_r$$ r -integral is not contained in the P$$_r$$ r -integral. Proceedings of the American Mathematical Society 150(5), 2107–2114 (2022)].