A worthy and celebrated extension is the Hörmander-Mihlin Theorem, which can be found in the first volume of  The Analysis of Linear Partial Differential Operators, Grundlehren 256, Springer-Verlag: let me call $g(D)$ your operator $T_g$.

If $g$ is a bounded function on $\mathbb R^n$
 such that  $ |x|^{k}|\nabla ^{k}g|  $is bounded for all $k\in [0, 1+n/2]$,
then the operator $g(D)$ is bounded on $L^p$ for $1<p<\infty$. Moreover, thanks to the Marcinkiewicz Theorem it is also bounded from $L^1$ to weak $L^1$
(also from the Hardy space $\mathscr H^1$ into $L^1$ and from $L^\infty$ to BMO).

Moreover if $g$ is a "multiplier" of the Schwartz space,
i.e. a smooth function increasing polynomially as well as all its derivatives, the operator $g(D)$ can be defined as an endomorphism of $\mathscr S'(\mathbb R^n)$. On the other hand, if $g$ is only bounded, $g(D)$ can be defined on $\mathscr S(\mathbb R^n)$ and extended as an endomorphism of $L^2(\mathbb R^n)$; above I give an example of what  happens if you control more derivatives : you obtain some $L^p$ boundedness. So the "maximal" extension to  $\mathscr S'(\mathbb R^n)$
requires full smoothness and temperate growth whereas $L^2$ extension can be done with $g$ only bounded and $L^p$ extension requires a specific behaviour for a finite number of derivatives.