Norm attaining bilinear forms of $$${\mathcal L}(^2 d_{*}(1, w)^2)$$$ at given vectors

Abstract

For given unit vectors $$$x_1, \cdots, x_n$$$ of a real Banach space $$$E,$$$ we define $$NA({\mathcal L}(^nE))(x_1, \cdots, x_n)=\{T\in {\mathcal L}(^nE): |T(x_1, \cdots, x_n)|=\|T\|=1\},$$ where $$${\mathcal L}(^nE)$$$ denotes the Banach space of all continuous $$$n$$$-linear forms on $$$E$$$ endowed with the norm $$$\|T\|=\sup_{\|x_k\|=1, 1\leq k\leq n}{|T(x_1, \ldots, x_n)|}$$$.
In this paper, we classify $$$NA({\mathcal L}(^2 d_{*}(1, w)^2))(Z_1, Z_2)$$$ for unit vectors $$$Z_1, Z_2\in d_{*}(1, w)^2,$$$ where $$$d_{*}(1, w)^2=\mathbb{R}^2$$$ with the norm of weight $$$0<w<1$$$ endowed with $$$\|(x, y)\|_{d_*(1, w)}=\max\Big\{|x|, |y|, \frac{|x|+|y|}{1+w}\Big\}$$$.