Journal of the Iranian Mathematical Society

Journal of the Iranian Mathematical Society

Geometry of variable-exponent Bochner-Lebesgue spaces‎: ‎Dentability‎, ‎Radon-Nikodym property‎, ‎and uniform convexity

Document Type : Dedicated to Prof. A. T.-M. Lau

Author
M. Yaremenko
National Technical University of Ukraine, “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine.
10.30504/jims.2026.567288.1310
Abstract
This paper provides a comprehensive study of geometric properties of variable-exponent Lebesgue-Bochner spaces $L^{p(\cdot)}(E‎, ‎X)$‎. ‎We establish that $L^{p(\cdot)}(E‎, ‎X)$ has the Radon-Nikodym property (RNP) if and only if $X$ does‎, ‎under the condition $p_m > 1$‎. ‎We investigate dentability of specific subsets‎, ‎showing that while the unit ball inherits dentability from $X$‎, ‎the set of simple functions may remain nondentable even when the space possesses RNP‎. ‎The perseverance of uniform convexity and smoothness are shown when both $X$ and $L^{p(\cdot)}(E)$ have these properties‎. ‎We introduce quantitative dentability moduli and relate the dentability modulus of $L^{p(\cdot)}(E,X)$ to that of $X$‎, ‎showing the relevance to the measure of $E$‎, ‎and the oscillation of $p(\cdot)$‎. ‎The paper reveals how variable exponents create hybrid geometries mixing $L^1$ and $L^p$ behaviors‎, ‎with implications for PDE theory‎, ‎optimization‎, ‎and geometric analysis‎.
Keywords
&lrm
Variable-exponent spaces&lrm
Bochner spaces
Radon-Nikodym property
dentability
Subjects
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Journal of the Iranian Mathematical Society
Volume 7, Issue 1
This issue is in progress but all papers are fully citable
February 2026
Pages 1-12