Henstock-Kurzweil Integral Transforms

We show conditions for the existence, continuity, and differentiability of functions defined by , where is a function of bounded variation on with . 1. Introduction Let be a complex function defined on a certain subset of . Many functions on functional analysis are integrals of the following form: We discuss the above function , where the integral that we use is that of Henstock-Kurzweil. This integral introduced independently by Kurzweil and Henstock in 1957-58 encompasses the Riemann and Lebesgue integrals, as well as the Riemann and Lebesgue improper integrals. In Lebesgue theory, there are well-known results about the existence, continuity, and differentiability of . For Henstock-Kurzweil integrals also there are results about this, for example, Theorems 12.12 and 12.13 of [1]. However, they all need the stronger condition: is bounded by a Henstock-Kurzweil integrable function . We provide other conditions for the existence, continuity, and differentiability of . 2. Preliminaries Let us begin by recalling the definition of Henstock-Kurzweil integral. For finite intervals in it is defined in the following way. Definition 2.1. Let be a function. One can say that is Henstock-Kurzweil (shortly, HK-) integrable, if there exists such that, for each , there is a function (named a gauge) with the property that for any -fine partition of (i.e., for each , ), one has The number is the integral of over and it is denoted as . In the unbounded case, the Henstock-Kurzweil integral is defined as follows. Definition 2.2. Given a gauge function , one can say that a tagged partition of is -fine, if (a)a = , ,(b) for all ,(c)？ . Definition 2.3. A function is Henstock-Kurzweil integrable on , if there exists such that, for each , there is a gauge for which (2.1) is satisfied for all tagged partition which is -fine according to Definition 2.2. Let be a function defined on an infinite interval , One can suppose that is defined on assuming that . Thus, is Henstock-Kurzweil integrable on if extended on is HK-integrable. For functions defined over intervals and One can makes similar considerations. Let be a finite or infinite interval. The space of all Henstock-Kurzweil integrable functions over is denoted by . This space will be considered with the Alexiewicz seminorm, which it is defined as follows: where the supremum is being taken over all intervals contained in . Definition 2.4. Let be a function, where is a finite interval. The variation of over the interval is defined as follows: We say that the function is of bounded variation on if . Now if is a function defined on