Speaker:
Francesco Tulone (Università degli Studi di Palermo, Italy)

Title:
From Zygmund’s ideas to HKr and Pr integrals

Abstract:
At the beginning of the 1900s Denjoy and Perron developed 
descriptive processes for recovering a function from its 
derivative that solved known problems of classical Riemann 
and Lebesgue integrals. Many years later an equivalent 
constructive Riemann-type integral process was developed 
by Henstock and Kurzweil. Both integration processes were 
generalized quite recently for many different spaces solving 
the problem of recovering Fourier coefficients in Haar, Walsh 
and Vilenkin systems. Many properties of these non-absolute 
integrals were investigated, for example the Hake property 
was studied with an abstract differential basis, in a 
topological spaces, in terms of variational measure and in 
Riesz spaces. In 1961 Calderon and Zygmund introduced the 
$L^r$-derivative to establish pointwise estimates for 
solutions of elliptic partial differential equations. In 1968 
L. Gordon described a Perron-type integral, the $P_r$-integral, 
that recovers a function from its $L^r$-derivative. 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 and it is 
an extension of the $P_r$-integral. In my talk I will show the 
recent results that I have obtained with Musial on 
$HK_r$-integral. We studied the integration by parts formula 
for the $HK_r$-integral, we described a norm on the space of 
$HK_r$-integrable functions and we investigated the dual and 
completion of this space. We discovered that, as in classical 
case the Henstock-Kurzweil integral is equivalent to the 
variational integral, also the for $HK_r$-integral we can 
define the $L^r$-variational integral and prove that it is 
equivalent to the $HK_r$-integral. Nevertheless,  comparing 
$HK_r$-integral and $P_r$-integral we showed that the first 
one is strictly wider than the second giving an unexpected 
result in comparison with the classical case and many other 
Perron-type and Henstock-Kurzweil-type integrals that were 
proved to be equivalent.