The Radon-Nikodym Theory for Banach Spaces Involving Variable Exponents

This paper establishes a comprehensive Radon-Nikodym theory for Banach space-valued functions within the framework of variable exponent Lebesgue spaces. We begin by developing a generalized Bochner integration theory for functions with respect to vector measures, leveraging a continuous bilinear mapping \(\Lambda : X \times Y \to Z\). A central result is the establishment of a fundamental duality: the space of bounded linear operators on the space of totally measurable functions is isometrically isomorphic to the space of vector measures with finite semi-variation.We then extend this framework to variable exponent Bochner spaces \(L^{p(\cdot)}(E, X, \Sigma, \mu)\). We prove that if \(p\in P\left(E\right)\), \(۱۱) for each \(\mu\)-measurable set \(D\subset E\), we obtain \(۱_{D} \left\| \psi \right\| \in L^{q\left(\cdot \right)} \left(E, \Sigma , \mu \right)\) and \(\left\| \psi ۱_{D} \right\| _{L^{q\left(\cdot \right)} \left(E, \Sigma , \mu \right)} =\left\| U_{D} \right\| _{L^{p\left(\cdot \right)} }\);۲) for each function \(f\in L^{p\left(\cdot \right)} \left(E, \Sigma , \mu \right)\) and each \(z^{*} \in Z^{*}\), the identity \(\left\langle U\left(f\right), z^{*} \right\rangle =\int _{D}\left\langle \psi \left(x\right)\left(f\left(x\right)\right), z^{*} \right\rangle d\mu \left(x\right),\) holds for all \(\mu\)-measurable set \(D\subset E\).

کلیدواژه ها:

Radon-Nikodym Property ، Variable Exponent Lebesgue Spaces ، Bochner Integral ، Vector Measures ، Dentability ، Operator-Valued Functions