The following definition of a short-time Fourier Transform (STFT) is well known in the literature: let $f,g \in L^2(\Bbb R^d)$ be two functions. The short-time Fourier Transform of $f$ w.r.t $g$ is given by
$$ V_gf(x,\xi) := (f \, T_x\bar g)\widehat{}(\xi) = \int_{\Bbb R^d } f(t)\overline{g(t-x)}e^{-2\pi i t \cdot \xi} \, dt, $$ for $x,\xi \in \Bbb R^d$.
My question is somewhat simpler, but derives from this definition. In fact, my question can be resumed to the usual Fourier Transform. More precisely, I am getting very confused about the expression below:
$$ (V_gf \, \overline{V_u h})\widehat{}(x,\xi). $$
In simple terms, this should be the usual Fourier Transform of the product between the two functions in parentheses. This is causing me trouble because I can't write out this expression in terms of integrals explicitily. I believe my problem comes from the understanding of what exactly is a Fourier transform of a function in two variables.
I obviously know that given a function $f$, we define its Fourier Transform by
$$ \hat{f}(w) = \int_{\Bbb R^d} f(x)e^{-2\pi i x \cdot w} dx, $$ for all $w \in \Bbb R^d$. I don't know how to extend this to the $2$-D case and I believe that's my only issue understand the problem presented above.
Thanks for any help in advance.