On $\ell$-weakly cross $t$-intersecting families for sets and vector spaces

Let $[n]$ (resp. $V$) be an $n$-element set (resp. $n$-dimensional vector space over the finite field $\mathbb{F}_{q}$), and $\binom{[n]}{k}$ (resp. $\genfrac{[}{]}{0pt}{}{V}{k}$) denote the set of all $k$-subsets of $[n]$ (resp. $k$-dimensional subspaces of $V$). We say that $\mathcal{F}\subseteq\binom{[n]}{k}$ (resp. $\mathcal{F}\subseteq \genfrac{[}{]}{0pt}{}{V}{k}$) and $\mathcal{G}\subseteq \binom{[n]}{k'}$ (resp. $\mathcal{G}\subseteq \genfrac{[}{]}{0pt}{}{V}{k'}$) are $\ell$-weakly cross $t$-intersecting if $\sum_{1\leq i,j\leq \ell}|F_{i}\cap G_{j}|\geq \ell^{2}t-\ell+1$ (resp. $\sum_{1\leq i,j\leq \ell}\dim(F_{i}\cap G_{j})\geq \ell^{2}t-\ell+1$) for all distinct $F_{1},\ldots,F_{\ell}\in\mathcal{F}$ and $G_{1},\ldots,G_{\ell}\in\mathcal{G}$. In this paper, we provide an alternative proof of the set version of the $\ell$-weakly cross $t$-intersecting theorem and an explicit lower bound for $n$. Moreover, we prove that if $\mathcal{F}$ and $\mathcal{G}$ are $\ell$-weakly cross $t$-intersecting subspace families, then $$|\mathcal{F}| \cdot |\mathcal{G}| \leq\genfrac{[}{]}{0pt}{}{n-t}{k-t}\genfrac{[}{]}{0pt}{}{n-t}{k'-t}$$ holds, provided that $n\geq (2k-t+1)(t+1)+(k-t+1)k'+k+2\ell-1$. This extends the theorem of Cao, Lu, Lv and Wang [J. Combin. Theory Ser. A 193 (2023), 105688], who established the upper bound for the product of the sizes of cross $t$-intersecting subspace families.