Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Reduced functions and Jensen measures

Authors: Wolfhard Hansen and Ivan Netuka
Journal: Proc. Amer. Math. Soc. 146 (2018), 153-160
MSC (2010): Primary 31B05, 31D05, 35J15, 60J45, 60J60, 60J75
Published electronically: July 10, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \varphi $ be a locally upper bounded Borel measurable function on a Greenian open set $ \Omega $ in $ \mathbb{R}^d$ and, for every $ x\in \Omega $, let $ v_\varphi (x)$ denote the infimum of the integrals of $ \varphi $ with respect to Jensen measures for $ x$ on $ \Omega $. Twenty years ago, B.J.Cole and T.J.Ransford proved that $ v_\varphi $ is the supremum of all subharmonic minorants of $ \varphi $ on $ X$ and that the sets $ \{v_\varphi <t\}$,
$ t\in \mathbf {R}$, are analytic. In this paper, a different method leading to the inf-sup-result establishes at the same time that, in fact, $ v_\varphi $ is the minimum of $ \varphi $ and a subharmonic function, and hence Borel measurable. This is presented in the generality of harmonic spaces, where semipolar sets are polar, and the key tools are measurability results for reduced functions on balayage spaces which are of independent interest.

References [Enhancements On Off] (What's this?)

  • [1] Mohammad Alakhrass and Wolfhard Hansen, Infima of superharmonic functions, Ark. Mat. 50 (2012), no. 2, 231-235. MR 2961319,
  • [2] Heinz Bauer, Harmonische Räume und ihre Potentialtheorie, Ausarbeitung einer im Sommersemester 1965 an der Universität Hamburg gehaltenen Vorlesung. Lecture Notes in Mathematics, No. 22, Springer-Verlag, Berlin-New York, 1966 (German). MR 0210916
  • [3] J. Bliedtner and W. Hansen, Simplicial cones in potential theory. II. Approximation theorems, Invent. Math. 46 (1978), no. 3, 255-275. MR 0492345,
  • [4] J. Bliedtner and W. Hansen, Potential theory, An analytic and probabilistic approach to balayage, Universitext, Springer-Verlag, Berlin, 1986. MR 850715
  • [5] B. J. Cole and T. J. Ransford, Subharmonicity without upper semicontinuity, J. Funct. Anal. 147 (1997), no. 2, 420-442. MR 1454488,
  • [6] Corneliu Constantinescu and Aurel Cornea, Potential theory on harmonic spaces, with a preface by H. Bauer; Die Grundlehren der mathematischen Wissenschaften, Band 158, Springer-Verlag, New York-Heidelberg, 1972. MR 0419799
  • [7] Stephen J. Gardiner, Harmonic approximation, London Mathematical Society Lecture Note Series, vol. 221, Cambridge University Press, Cambridge, 1995. MR 1342298
  • [8] Stephen J. Gardiner, Myron Goldstein, and Kohur GowriSankaran, Global approximation in harmonic spaces, Proc. Amer. Math. Soc. 122 (1994), no. 1, 213-221. MR 1203986,
  • [9] Alexander Grigor'yan and Wolfhard Hansen, A Liouville property for Schrödinger operators, Math. Ann. 312 (1998), no. 4, 659-716. MR 1660247,
  • [10] W.Hansen.
    Three views on potential theory,
    a course at Charles University (Prague), Spring 2008.
  • [11] Wolfhard Hansen and Ivan Netuka, Jensen measures in potential theory, Potential Anal. 37 (2012), no. 1, 79-90. MR 2928239,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 31B05, 31D05, 35J15, 60J45, 60J60, 60J75

Retrieve articles in all journals with MSC (2010): 31B05, 31D05, 35J15, 60J45, 60J60, 60J75

Additional Information

Wolfhard Hansen
Affiliation: Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

Ivan Netuka
Affiliation: Faculty of Mathematics and Physics, Charles University, Mathematical Institute, Sokolovská 83, 186 75 Praha 8, Czech Republic

Keywords: Reduced function, Jensen measure, axiom of polarity
Received by editor(s): November 5, 2016
Received by editor(s) in revised form: January 18, 2017
Published electronically: July 10, 2017
Communicated by: Zhen-Qing Chen
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society