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

Abstract Given arbitrary $$r\ge 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , we construct an HK $$_r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mi>r</mml:mi> <mml:mrow/> </mml:mmultiscripts> </mml:math> -integrable function which is not P $$_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mn>1</mml:mn> <mml:mrow/> </mml:mmultiscripts> </mml:math> -integrable. This is an extension of a recently published construction [Musial, P., Skvortsov, V., Tulone, F.: The HK $$_r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mi>r</mml:mi> <mml:mrow/> </mml:mmultiscripts> </mml:math> -integral is not contained in the P $$_r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mi>r</mml:mi> <mml:mrow/> </mml:mmultiscripts> </mml:math> -integral. Proceedings of the American Mathematical Society 150 (5), 2107–2114 (2022)].