The L^rHenstock–Kurzweil integral

Abstract

Abstract. We present a method of integration along the lines of the HenstockKurzweil integral. All L r -derivatives are integrable in this method.

The $$L^{r}$$-Variational Integral

“…To establish pointwise estimates for solutions of elliptic partial differential equations, in 1961 Calderon and Zygmund introduced the L r -derivative (see [3]) and in 1968 L. Gordon described a Perron-type integral, the P rintegral, that recovers a function from its L r -derivative (see [4]). In 2004, Musial and Sagher extended the P r -integral to the L r -Henstock-Kurzweil integral, the HK r -integral, that recovers also a function from its L r -derivative (see [6]). Quite recently the integration by parts formula for the HK r -integral was investigated by Musial and Tulone (see [7]) and the same authors described a norm on the space of HK r -integrable functions and studied the dual and completion of this space (see [8]).…”

“…The Henstock-Kurzweil integral primitive is continuous in the usual sense. In [6] is proved an equivalent result for L r -Henstock-Kurzweil indefinite integral.…”

“…By Theorem 5 in [6], the function F in the Definition 2.2 is unique up to an additive constant, so we can state that for each x ∈ (a, b]…”

On Descriptive Characterizations of an Integral Recovering a Function from Its $$L^r$$-Derivative

“…The HK r -integral turned out to be strictly wider than the P r -integral (see [7]). It was also shown in [6] that the indefinite HK r -integral is L r -differentiable almost everywhere and belongs to a Lusin-type class of ACG r -functions. Some other properties of these integrals are investigated in [9,10,11,12].…”

“…In this paper we obtain a full descriptive characterization of a Henstock-Kurzweil-type integral, the HK r -integral (defined in 2004 by Musial and Sagher [6]), which serves to integrate a derivative defined in the space L r , 1 ≤ r < ∞. This L r -derivative was introduced in [1] by Calderón and Zygmund to be used in some estimates for solutions of elliptic partial differential equations.…”