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The Hörmander multiplier theorem I: The Linear Case

We discuss $L^p(\mathbb R^n)$ boundedness for Fourier multiplier operators that satisfy the hypotheses of the Hörmander 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 $|\frac 1p-\frac12|=\frac sn$.