On Carleson measures and Hankel forms of higher weights on Hardy space

In this book we are concerned with two problems. The first problem is to describe Carleson measures for Hardy Sobolev spaces and the generalized Bergman spaces in the unit ball. We give by means of Green’s formula an alternative proof of the characterization of Carleson measures for some Hardy Sobolev spaces (including Hardy space). We also give a new characterization of Carleson measures for the generalized Bergman spaces (including the Drury-Averson space) on the unit ball in terms of testing on balls using T(1)-Theorem type techniques for singular integrals. The second problem concerns bilinear Hankel forms of higher weights on Hardy spaces of the unit ball. We show that these Hankel forms are bounded and compact if and only if the symbols satisfy certain Carleson measure criteria and vanishing Carleson measure criteria respectively. We get a full characterization of Schatten class Sp Hankel forms for p great or equal to 2 in terms of the membership for the symbols to be in certain Besov spaces.