Let $x,y \in \mathbb R^d,$ and $0\neq t \in \mathbb R.$
Define $f(y)= \sup_{x\in \mathbb R^d}\{e^{-\pi |y-tx|^2/ (1+t^2)} \}.$
My Question is: Is it true that $f\in L^{1}(\mathbb R^d)$? Is it true that $\sup_{x\in \mathbb R^d} \int_{\mathbb R^d} |e^{-\pi |y-tx|^2/ (1+t^2)} | dy < \infty $ ? If yes/no, how to justify it?
[I am familiar with the fact that: $\int_{\mathbb R^d} e^{-b x^2} dx =b^{-d/2}$, $(b>0)$]
Let us look at $f$, for every $y$, as inside the $sup$, you have the exponential of a negative number, it is lesser than 1 or equal, and continuous. 1 is reached by taking $x=\frac{y}{t}$, then, for all $y$, $f(y)=1$. So your first assumption is incorrect.
For the second part, you can, for all $x$, work in a finite variable-sized ball/cube, change the variable to get something you're more familiar with, take the limit, and see what you end up with.