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

Abstract

It is proved that any function of a Lusin-type class, the class of ACGr-functions, is differentiable almost everywhere in the sense of a derivative defined in the space L r , 1 ≤ r < ∞. This leads to obtaining a full descriptive characterization of a Henstock-Kurzweil-type integral, the HKrintegral, which serves to recover functions from their L rderivatives. The class ACGr is compared with the classical Lusin class ACG and it is shown that a continuous ACGfunction can fail to be L r -differentiable almost everywhere.

Comparison of the P-integral with Burkill's integrals and some applications to trigonometric series

“…Then Proposition 2.17 implies that f and all considered functions are SCP -integrable on any compact interval, in particular on [0, 2π], with respect to the basis coinciding with the whole interval. This means that in our case we can put β = 0 in formulas (12) and (13). Hence, by the consistency of the P r -, CP -and SCP -integrals, formulas (12) and (13) imply formulas ( 14) and (15) giving the desired result.…”

“…Originally a version of this L r -derivative appeared in an earlier paper [7] by Calderon and Zygmund and was used to obtain some estimates for solutions of elliptic partial differential equations. Note that the problem of recovering a function from its L r -derivative can be solved also by a Kurzweil-Henstock-type integral, the HK r -integral, which was introduced by Musial and Sagher in [11] (see its equivalent definitions in [13] and in [15]) and which turned out to strictly include the P r -integral (see [12]). In [14] Musial and Tulone developed a dual to the space of HK r -integrable functions.…”