Given an operator $Q$ between a Hilbert space $U$ and $L^2(ℝ^d;ℝ^d)$, is it possible to make sense of $U∋u↦(Qu)(x)$ for a fixed $x∈ℝ^d$?

Question

Let

• $U$ be a Hilbert space
• $H:=L^2(\mathbb R^d;\mathbb R^d)$ for some $d\in\left\{2,3\right\}$
• $Q$ be a Hilbert-Schmidt operator from $U$ to $H$.

I want that $\tilde Q(x)$, where $$\tilde Q(x):=U\ni u\mapsto (Qu)(x)\;\;\;\text{for }x\in\mathbb R^d\;,\tag 1$$ is a Hilbert-Schmidt operator from $U$ to $\mathbb R^d$, for all $x\in\mathbb R^d$.

Clearly, at the moment, $(1)$ is not even well-defined, cause we cannot talk about the value of a function belonging to $H$ at a single point.

So, my question is: Is it possible to make sense of $(1)$? Maybe by choosing $H$ to be a suitable Sobolev space?