Some remarks on Vainikko integral operators in BV type spaces
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Some remarks on Vainikko integral operators in BV type spaces Laura Angeloni1
· Jürgen Appell2 · Simon Reinwand3
Ricordando Mimmo, collega e amico, con affetto, stima e gratitudine Received: 31 May 2020 / Accepted: 10 July 2020 © The Author(s) 2020
Abstract In this paper we study Vainikko integral operators which are similar to so-called cordial integral operators and contain the classical Hardy operator, the Schur operator, and the Hilbert transform as special cases. For such operators we obtain norm estimates and equalities, mainly in BV type spaces in the sense of Jordan, Wiener, Riesz, and Waterman. Several examples are also discussed. Keywords Cordial integral operator · Vainikko integral operator · BV type space · Norm estimate Mathematics Subject Classification 47G10 · 26A45 · 45D05 · 45H05 · 45P05
The first author is a member of the group GNAMPA of the Istituto Nazionale di Alta Matematica (INdAM) and of the “Research Italian Network on Approximation”; she was partially supported by Department of Mathematics and Computer Science of the University of Perugia (Italy), by the project “Integrazione, Approssimazione, Analisi Nonlineare e loro Applicazioni”, funded by the 2019 basic research fund of the University of Perugia and by the 2020 GNAMPA project “Metodi di analisi reale e di teoria della misura per l’approssimazione attraverso operatori discreti e applicazioni”.
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Laura Angeloni [email protected] Jürgen Appell [email protected] Simon Reinwand [email protected]
1
Department of Mathematics and Computer Science, University of Perugia, Via Vanvitelli 1, 06123 Perugia, Italy
2
Department of Mathematics, University of Würzburg, Emil-Fischer-Str. 30, 97074 Würzburg, Germany
3
Department of Mathematics, University of Würzburg, Emil-Fischer-Str. 40, 97074 Würzburg, Germany
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1 Introduction Domenico Candeloro (called “Mimmo” by his friends) was one of the leading specialists in the theory, methods, and applications of integral operators. He has given many important contributions to this field, with a particular emphasis on exotic measures, non-standard integrals, and multivalued maps. In the list of references at the end we mention only the more recent papers [2–25] he wrote, in part with coauthors, in the last 20 years. In this paper we study a class of integral operators in a much simpler setting, using only single-valued scalar functions and integrals defined by the classical Lebesgue measure on the real line. In spite of their simplicity, we are convinced that our results on mapping properties of such integral operators would have been appreciated by Mimmo. In our discussion we will give particular attention to spaces of functions of bounded variation, a topic that is also very much en vogue in the Analysis School of the University of Perugia which owes so much to Mimmo’s scientific activity.
2 Cordial integral operators Given a nonnegative L 1 function ϕ : (0, 1) → R, in [27] the author defines an associated Volterra integral operator Vϕ by 1 (Vϕ x)(t) = t
t 0
ϕ
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