THE DISTRIBUTION FUNCTION INEQUALITY AND PRODUCTS OF TOEPLITZ-OPERATORS AND HANKEL-OPERATORS

In this paper we study Hankel operators and Toeplitz operators through a distribution function inequality on the Lusin area integral functio n and the Littlewood-Paley theory. A sufficient condition and a necess ary condition are obtained for the boundedness of the product of two H ankel operators. They lead to a way to approach Sarason's conjecture o n products of Toeplitz operators and shed light on the compactness of the product of Hankel operators. An elementary necessary and sufficien t condition for the product of two Toeplitz operators to be a compact perturbation of a Toeplitz operator is obtained. Moreover, a necessary condition is given for the product of Hankel operators to be in the c ommutator ideal of the algebra generated by the Toeplitz operators wit h symbols in a Sarason algebra. (C) 1996 Academic Press, Inc.