The 𝐻𝐾ᵣ-integral is not contained in the 𝑃ᵣ-integral

Abstract

We compare a Perron-type integral with a Henstock-Kurzweil-type integral, both having been introduced to recover functions from their generalized derivatives defined in the metric L r L^r . We give an example of an H K r HK_r -integrable function which is not P r P_r -integrable, thereby showing that the first integral is strictly wider than the second one.

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.…”

“…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.…”

“…We use below the following generalization of Theorem 4.2 which is a particular case of [5, Theorem 5.3 (i)], provided its proof is corrected according to [17] or [6]: Theorem 4.3. If the series (11) converges to a function f (x) almost everywhere and the partial sums of (11) are bounded for each x except an countable set then the conclusion of Theorem 4.2 holds true, in particular formulas (12) and ( 13) hold with the same meaning for β and B. Now we apply this theorem together with Theorem 3.3 to obtain Theorem 4.4.…”

“…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.…”

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

“…The HK r -integral was defined as an extension of a Perron-type integral, the P r -integral, which was defined earlier by L. Gordon [3] and which also recovers 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.…”