H1-integrals with respect to some bases

In 1998, Garces, Lee, and Zhao defined the so-called H (1)-integral [I.J.L. Garces, P.Y. Lee, and D. Zhao, Moore-Smith limits and the Henstock integral, Real Anal. Exch., 24(1):447-456, 1998/99]. Later on, a full characterization of H (1)-integrable functions was given by Maliszewski and the author [A. Maliszewski and P. Sworowski, A characterization of H (1)-integrable functions, Real Anal. Exch., 28(1):93-104, 2002/03]. The characterization is in terms of generalized continuity and so is similar to the classical Lebesgue theorem on Riemann integrands. The present paper contributes to the theory with some results of this type for H (1)-integral defined with respect to differential bases: namely, the McShane basis, path bases, and so-called free point bases. In the latter two cases, the characterization is partial (necessity condition being provided only).