Weak-strong convolution operators on certain disconnected groups

Garth Gaudey; Ian Inglis

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The purposes of this paper may be described as follows: (i) to provide a useful substitute for the Cotlar-Stein lemma for L^p-spaces (the orthogonality conditions are replaced by certain fairly weak smoothness asumptions); (ii) to investigate the gap between the Hörmander multiplier theorem and the Littman-McCarthy-Rivière example - just how little regularity is really needed? (iii) to simplify and extend the work of Duoandikoetxea...

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R. Fefferman has shown that, on a product-space with two factors, an operator T bounded on $L^{2}$ maps $L^{\infty}$ into BMO of the product if the mean oscillation on a rectangle R of the image of a bounded function supported out of a multiple R’ of R, is dominated by $C | R |^{s} | R |^{- s}$ , for some $s > 0$ . We show that this result does not extend in general to the case where E has three or more factors but remains true in this case if in addition T is a convolution operator, provided $s > s_{0} (E)$ . We also show that the Calderon-Coifman...