Theorems on the integrability of products of functions for the Kurzweil-Henstock integral

Interest in the Kurzweil-Henstock integral (the gauge integral) has been rising over the last few decades, since its definition is only slightly different from the definition of the Riemann integral; the class of Kurzweil-Henstock integrable functions contains functions that are Riemann integrable (in the proper and improper sense), and f is Lebesgue integrable if and only if f and |f| are Kurzweil-Henstock integrable functions. The Kurzweil-Henstock integral has the usual properties of linearity, additivity, and monotonicity, and it satisfies counterparts of some classical theorems, such as a Beppo Levi--type lemma, a Fatou-type lemma, a Lebesgue-type theorem, etc. par In this paper the question of the integrability of the product of integrable functions in the Kurzweil-Henstock sense is considered. The classical result is the theorem on the integrability of the product of an integrable function and a function of bounded variation. By a simple modification of the reasoning used in the proof of that classical result, the authors present the following result: Let f and g be functions on [a,b], such that f is integrable and g is bounded, and suppose that the Stieltjes integral S=int_{a}^bg(x)dF(x) exists, where F(x)=int_{a}^xf(t)dt; then fg is also integrable on [a,b] and its Kurzweil-Henstock integral is equal to S. In this article, this fact has a major role in establishing several more general statements on the integrability of the product of integrable functions in the Kurzweil-Henstock sense. par The first main result of the paper is the following statement for the Kurzweil-Henstock integral: If f is an integrable function and its primitive F satisfies the Hölder condition with exponent alpha and g satisfies the Hölder condition with exponent beta, where {alpha+beta> 1}, then fg is integrable. This result follows directly from the previous fact and the well-known theorem of V. T. Kondurar [Mat. Sb. (2) {bf 2} (1937), 361--366; JFM 63.0186.03] on the existence of the Stieltjes integral in the case when functions F and g satisfy the Hölder condition, respectively with exponents alpha and beta, and the inequality {alpha+ beta> 1} holds. Recall that the functions that satisfy the Hölder condition do not necessarily have bounded variation even in the case of their differentiability, and therefore the previous assertion is not contained in the theorem on the integrability of the product of an integrable function and a function of bounded variation. par Based on more general results [L.~C. Young, Acta Math. {bf 67} (1936), no.~1, 251--282; [msn] MR1555421 [/msn]; Proc. London Math. Soc. (2) {bf 43} (1937), no.~6, 449--467; [msn] MR1575654 [/msn]; R.~Leśniewicz and W. Orlicz, Studia Math. {bf 45} (1973), 71--109; [msn] MR0346509 [/msn]; P.~P. Lévy, {it Concrete problems of functional analysis} (Russian), With a supplement by F. Pellegrino on analytic functions, Russian translation, Izdat. "Nauka", Moscow, 1967; [msn] MR0223845 [/msn]], analogously, the second main result is obtained for functions, one of which has a primitive that satisfies the generalized Hölder condition with the modulus phi, and the second itself satisfies the generalized Hölder condition with the modulus psi, where the function tmapsto t^{-2}phi(t) psi (t) is integrable in a neighborhood of zero. par Finally, similar assertions have been established for functions with bounded variations in the sense of Wiener, Yang, Waterman, and Shram [J.~M. Appell, J. Banaś and N.~J. Merentes~Díaz, {it Bounded variation and around}, De Gruyter Ser. Nonlinear Anal. Appl., 17, De Gruyter, Berlin, 2014; [msn] MR3156940 [/msn]; L.~C. Young, op. cit.; [msn] MR1555421 [/msn]; [msn] MR1575654 [/msn]; R. Leśniewicz and W. Orlicz, op. cit.; [msn] MR0346509 [/msn]; P.~P. Lévy, op. cit.; [msn] MR0223845 [/msn]; M. Schramm, Trans. Amer. Math. Soc. {bf 287} (1985), no.~1, 49--63; [msn] MR0766206 [/msn]]