Two problems of Calderón-Zygmund theory on product-spaces
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 bicommutators, obtained from the Calderon commutators by multilinear tensor product, are bounded on \(L^ 2\) with norms growing polynomially.
scientific article; zbMATH DE number 4040629