Too long for a comment. 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 $L^1$ (and from $L^\infty$ to BMO).