BESOV-SPACES, MEAN-OSCILLATION, AND GENERALIZED HANKEL-OPERATORS

We introduce some operators on the Bergman space A2 on the unit ball that generalize the classical (big) Hankel operator. For such operators we prove boundedness, compactness, and Schatten-ideal property criteria. These extend known results. These new operators are defined in terms of a symbol. We prove in particular that for 2 less-than-or-equal-to p < infinity, these operators belong to the Schatten ideal S(p) if and only if the symbol f is in the Besov space B(p). We also give several different characterizations of the norm on the Besov spaces B(p). In particular we prove that the Besov spaces are the mean oscillation spaces in the Bergman metric, for 1 less-than-or-equal-to p < infinity.