On the strong subdifferentiability of the homogeneous polynomials and (symmetric) tensor products

In this paper, we study the (uniform) strong subdifferentiability of the norms of the Banach spaces $\mathcal{P}(^N X, Y^*)$, $X \hat{\otimes}_\pi \cdots \hat{\otimes}_\pi X$ and $\hat{\otimes}_{\pi_s,N} X$. Among other results, we characterize when the norms of the spaces $\mathcal{P}(^N \ell_p, \ell_{q}), \mathcal{P}(^N l_{M_1}, l_{M_2})$, and $\mathcal{P}(^N d(w,p), l_{M_2})$ are strongly subdifferentiable. Analogous results for multilinear mappings are also obtained. Since strong subdifferentiability of a dual space implies reflexivity, we improve some known results on the reflexivity of spaces of $N$-homogeneous polynomials and $N$-linear mappings. Concerning the projective (symmetric) tensor norms, we provide positive results on the subsets $U$ and $U_s$ of elementary tensors on the unit spheres of $X \hat{\otimes}_\pi \cdots \hat{\otimes}_\pi X$ and $\hat{\otimes}_{\pi_s,N} X$, respectively. Specifically, we prove that $\hat{\otimes}_{\pi_s,N} \ell_2$ and $\ell_2 \hat{\otimes}_\pi \cdots \hat{\otimes}_\pi \ell_2$ are uniformly strongly subdifferentiable on $U_s$ and $U$, respectively, and that $c_0 \hat{\otimes}_{\pi_s} c_0$ and $c_0 \hat{\otimes}_\pi c_0$ are strongly subdifferentiable on $U_s$ and $U$, respectively, in the complex case.