Geometry of Lebesgue-Bochner Function Spaces-Smoothness

Abstract

This paper contains a complete solution to the problem of the higher order differentiability of the norm function in the Lebesgue-Bochner function spaces L (E,y>) , 1 <1 p <. oo , where E is a real Banach and p-is an extended real-valued measure defined on the measurable space (T,£).

Fréchet differentiable points in Bochner function spaces 𝐿_{𝑝}(𝜇,𝑋)

“…Consequently, it follows that Lp(p,X) is Fréchet differentiable if and only if X is Fréchet differentiable. This result is due to Leonard and Sundaresan [2]. DEFINITION 1.…”

Rotundity in Lebesgue-Bochner function spaces

“…This question has an affirmative answer whenever X is reflexive. This follows from the now familiar duality and a theorem of Leonard and Sundaresan [12] stating that if 1 <p < oo, X and Lp(fi, X) have or fail to have Fréchet differentiable norms together. With this as supporting evidence along with the fact that the combination of rotundity and the Radon-Riesz property is a condition lying in strength between local uniform rotundity and rotundity, each of which lifts from X to Lp( /¿, X), an affirmative answer to this question might be expected.…”

Renormings of L p (L q )

“…We improve the results in [11] in the following sense: if 1 < r ≤ p and the space X admits a norm which is C n -smooth, where n is the largest integer strictly less than r and such that its n-th derivative is (r − n)-Hölder on the unit sphere, then the norm on the space L p (X) has the same properties of differentiability.…”