An HK$_r$-integrable function which is P$_s$-integrable for no $s$

Abstract

Given arbitrary $r\ge1$, we construct an HK$_r$-integrable function which is not P$_1$-integrable. This is an extension of Musial et al.\ construction published recently in [Musial, P., Skvortsov, V., Tulone, F.: The HK$_r$-integral is not contained in the P$_r$-integral. Proc. Amer. Math. Soc. {\bf150}(5), 2107--2114 (2022)].