  We discuss $L^p(\mathbb R^n)$ boundedness for Fourier multiplier operators
that satisfy the hypotheses of the H\"ormander multiplier theorem in terms of
an optimal condition that relates the distance $|\frac 1p-\frac12|$ to the
smoothness $s$ of the associated multiplier measured in some Sobolev norm. We
provide new counterexamples to justify the optimality of the condition $|\frac
1p-\frac12|<\frac sn$ and we discuss the endpoint case $|\frac
1p-\frac12|=\frac sn$.
