On $\mathrm{\ell }$-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 $\left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ (resp. $\left[\genfrac{}{}{0px}{}{V}{k}\right]$) denote the set of all $k$-subsets of $[n]$ (resp. $k$-dimensional subspaces of $V$). We say that $\mathcal{F}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ (resp. $\mathcal{F}\subseteq \left[\genfrac{}{}{0px}{}{V}{k}\right]$) and $\mathcal{G}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{k^{\mathrm{\prime }}}\right)$ (resp. $\mathcal{G}\subseteq \left[\genfrac{}{}{0px}{}{V}{k^{\mathrm{\prime }}}\right]$) are $\mathrm{\ell }$-weakly cross $t$-intersecting if ${\sum }_{1\le i,j\le \mathrm{\ell }}\mathrm{\mid }{F}_i\cap G_j\mathrm{\mid }\ge {\mathrm{\ell }}^{2}t-\mathrm{\ell }+1$ (resp. ${\sum }_{1\le i,j\le \mathrm{\ell }}\dim (F_i\cap G_j)\ge {\mathrm{\ell }}^{2}t-\mathrm{\ell }+1$) for all distinct $F_1,\dots ,F_{\mathrm{\ell }}\in \mathcal{F}$ and $G_1,\dots ,G_{\mathrm{\ell }}\in \mathcal{G}$. In this paper, we provide an alternative proof of the set version of the $\mathrm{\ell }$-weakly cross $t$-intersecting theorem and an explicit lower bound for $n$. Moreover, we prove that if $\mathcal{F}$ and $\mathcal{G}$ are $\mathrm{\ell }$-weakly cross $t$-intersecting subspace families, then $$\mathrm{\mid }\mathcal{F}\mathrm{\mid }\cdot \mathrm{\mid }\mathcal{G}\mathrm{\mid }\le \left[\genfrac{}{}{0px}{}{n-t}{k-t}\right]\left[\genfrac{}{}{0px}{}{n-t}{k^{\mathrm{\prime }}-t}\right]$$ holds, provided that $n\ge (2k-t+1)(t+1)+(k-t+1)k^{\mathrm{\prime }}+k+2\mathrm{\ell }-1$. 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.