This 1995 paper contains a sharp version of the classical Marcinkiewicz multiplier theorem for the class of homogeneous Fourier multipliers in two dimensions; here a one-dimensional Marcinkiewicz condition is sufficient. Examples are given to show that the straightforward extension of this statement to higher dimensions does not hold. We provide appropriate versions in higher dimensions which rely on a Fefferman-Stein weighted norm inequality for a relevant C��rdoba type square function. The proof of this inequality is given by an early version of an induction on scales argument. For $p<1$ we prove a multiplier theorem on product-type $H^p$-spaces.

28 pages. The online version of the journal issue provided by the current publisher has faulty typesetting. A 1995 posting on arXiv shows only 7 pages. We therefore post the original preprint version of the paper