Following Bernstein-Zelevinski, an $\ell$-space is a Hausdorff, locally compact totally disconnected topological space. For an $\ell$-space $X$, denote $S(X)$ the space of Bruhat-Schwartz functions on $X$, i.e., the space of $\mathbb{C}$-valued locally constant, compact supported functions on $X$. A distribution on $X$ is defined to be an element in $\textrm{Hom}_{\mathbb{C}}(S(X),\mathbb{C})$.

Let $F$ be a $p$-adic field.

$\textbf{Question 1}$: Let $G=F^\times$ and $X=F^2-\{(0,0)\}$. Let $G$ act on $X$ by $a.(x,y)=(ax,ay),a\in F^\times, (x,y)\in X$. Let $U=\{(x,y)\in X, xy\ne 0\}$. Is there a $G$-invariant distribution $T$ on $X$ such that the restriction of $T$ to $U$ is non-zero?

$\textbf{Question 1'}$: Let $G,X,U$ be as in $\textbf{Question 1}$. Let $\psi$ be a non-trivial additive character of $F$ and let $\mu=\psi(y/x)d^*xd^*y$, viewed as a measure on $U$. It is clear that $\mu$ is $G$-invariant. Is there a $G$-invariant distribution on $X$ such that $T|_U=\mu$?

$\textbf{Question 2}$: Let $G=F^\times \times F^\times$ and $X=F^3-\{(0,0,0)\}$. Let $U=(F^\times)^2$. Let $G$ act on $X$ by $(a,b).(x,y,z)=(ax,by,abz)$. Is there a $G$-invariant distribution on $X$ such that its restriction to $U$ is non-trivial?

$\textbf{Question 2'}$: Let $G,X,U$ be as in $\textbf{Question 2}$. Let $\mu=\psi(z/(xy))d^*xd^*yd^*z$, viewed as a distribution on $U$. Here $\psi$ is still taken as an additive character of $F$. Is there a $G$-invariant distribution $T$ on $X$ such that $T|_U=\mu$.

Here is one example in my mind that suggests the non-existence of distributions $T$ in Question 1 and 2. Consider the multiplicative action of $F^\times$ on $F$, and let $U=F^\times$, which is an open subset of $F$. Then by Tate's thesis, there is no $F^\times$-invariant distribution $T$ on $F$ such that $T|_U$ is non-zero. In fact, the only $F^\times$-invariant distribution on $F$ is the Dirac measure.

But if we consider the action $a.(x,y)=(ax,a^{-1}y)$ in $\textbf{Question 1}$, there clearly exists $G$-invariant distribution on $X$ (even on $F^2$, say the usual Haar measure $dxdy$ on $F^2$) such that its restriction to $U$ is non-trivial.

Any thoughts or suggestions? Thanks in advance.

I asked part of the question here https://math.stackexchange.com/questions/2409933/extension-of-certain-invariant-distributions, but did not draw much attention.