Abstract
We prove pointwise convexity (Jensen-type) inequalities of the form
where F is a convex function defined on a convex subset of some Banach space X and T is the X-valued extension of a positive operator on some function space. Examples include the pointwise Hölder inequality T(fg) ≤ (Tfp)1/p (Tfq)1/q for a positive sublinear operator T. As applications we consider vector-valued conditional expectation and a ``real'' proof of the Riesz-Thorin theorem for positive operators.
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References
Y. A. Abramovich, C. D. Aliprantis, An invitation to operator theory, volume 50 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2002.
Colin Bennett, Robert Sharpley, Interpolation of operators, volume 129 of Pure and Applied Mathematics, Academic Press Inc., Boston, MA, 1988.
Nicolas Bourbaki. Integration, I. Chapters 1–6. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2004. Translated from the 1959, 1965 and 1967 French originals by Sterling K. Berberian.
J. Diestel, J.J.jun. Uhl, Vector Measures, Mathematical Surveys. No.15. Providence, R.I.: American Mathematical Society. XIII, 322 p. 1977.
Nelson Dunford, Jacob T. Schwartz, Linear Operators. I. General Theory. (Pure and Applied Mathematics. Vol. 6) New York and London: Interscience Publishers. XIV, 858 p. 1958.
Ivar Ekeland, Roger Temam, Convex analysis and variational problems, North-Holland Publishing Co., Amsterdam, 1976. Translated from the French, Studies in Mathematics and its Applications, Vol. 1.
Loukas Grafakos, Classical and Modern Fourier Analysis, Prentice Hall, 870 p. 2004.
S. G. Krein, Yu. Ī. Petunīn, E. M. Semënov, Interpolation of linear operators, volume 54 of Translations of Mathematical Monographs, American Mathematical Society, Providence, R.I., 1982. Translated from the Russian by J. Szűcs.
Ulrich Krengel, Ergodic theorems, volume 6 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel.
Lech Maligranda, Why Hölder's inequality should be called Rogers' inequality, Math. Inequal. Appl., 1(1):69–83, 1998.
Lech Maligranda, Positive bilinear operators in Calderón-Lozanovskii spaces, Arch. Math. (Basel), 81(1):26–37, 2003.
Helmut H. Schaefer, Banach Lattices and Positive Operators, Die Grundlehren der mathematischen Wissenschaften. Band 215. Berlin-Heidelberg-New York: Springer-Verlag. XI, 376 p. 1974.
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Haase, M. Convexity Inequalities for Positive Operators. Positivity 11, 57–68 (2007). https://doi.org/10.1007/s11117-006-1975-4
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DOI: https://doi.org/10.1007/s11117-006-1975-4