Let $X$ be the space of functions on $\mathbb R^d \times \mathbb R^d$ $(d\in \mathbb N),$ that is, $X=\{f:\mathbb R^{d}\times \mathbb R^d \to \mathbb C: f \text{is function} \}.$ Assume that $(X, \|\cdot\|_{X(\mathbb R^{2d})})$ is a Banach algebra. Therefore, we have $\|fg\|_{X(\mathbb R^{2d})} \leq C \|f\|_{X(\mathbb R^{2d})} \|g\|_{X(\mathbb R^{2d})}$ for all $f, g \in X.$
Let $G\in X $ so $G:\mathbb R^{d}\times \mathbb R^d \to \mathbb R: (x,y)\mapsto G(x,y)\in \mathbb R$ be a function. Define $g(x,y)= G(x,x)$ for all $x,y \in \mathbb R^{d}.$ Roughly speaking $g$ is a function on $\mathbb R^d.$
Question: Can we expect $\|fg\|_{X(\mathbb R^{2d} )} \leq C \|f\|_{X(\mathbb R^{2d})} \|g\|_{X(\mathbb R^d)}$? Specifically, is there any examples of Banach spaces $X-$ where one can expect this kind of scenario? (e.g., Sobolev spaces or so..)
Side Thoughts: Define $h:\mathbb R^{2d} \to \mathbb R$ such $h(x,y)= h(x,x)$ for all $x,y \in \mathbb R^d.$ Can we say $\int_{\mathbb R^{2d}} |h(x,y)| dx dy = \int_{\mathbb R^d} |h(x,x)| dx$?