Weak$^*$-weak points of continuity and very smooth points in Bochner function spaces $L^{p}(μ, X)$

Abstract: Suppose that $([0,1],\mathcal{A},μ)$ is the Lebesgue measure space and $1&lt; p&lt;\infty$ with $\frac{1}{p}+\frac{1}{q}=1$. We prove that an unit vector $h\in L^{p}(μ, X)<sup>*$ is a weak$<sup>*$-weak point of continuity of $S(L^{p}(μ, X)<sup>*)$ if and only if $\frac{h(t)}{|h(t)|}$ is a weak$<sup>*$-weak point of continuity of $S(X<sup>*)$ for almost every $t\in [0, 1]$ with $h(t)\neq 0$, where $X$ is a Banach space such that $X<sup>*$ is separable. As an application of this result, under the same hypothesis on $X$, we prove that a point $f\in L^p(μ, X)_{1}$ is a very smooth point if and only if $\frac{f(t)}{|f(t)|}\in S(X)$ is a very smooth point for almost every $t\in [0, 1]$ with $f(t)\neq 0$. We give several examples to show that these results does not hold for $p=1$.