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 function and the Littlewood–Paley theory. A sufficient condition and a necessary condition are obtained for the boundedness of the product of two Hankel operators. They lead to a way to approach Sarasonʹs conjecture on products of Toeplitz operators and shed light on the compactness of the product of Hankel operators. An elementary necessary and sufficient 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 commutator ideal of the algebra generated by the Toeplitz operators with symbols in a Sarason algebra.