LAUSR

We discuss L(R) boundedness for Fourier multiplier operators that satisfy the hypotheses of the Hörmander multiplier theorem in terms of an optimal condition that relates the distance | 1 p − 1 2 | to the smoothness s of the associated multiplier measured in some Sobolev norm. We provide new counterexamples to justify the optimality of the condition | 1 p − 1 2 | < s n and we discuss the endpoint case | 1 p − 1 2 | = s n .