On the Boundedness and Compactness of Operators of Hankel Type

Commutators on Half-Spaces

“…Our result is not entirely new. The following theorem was due to Beatrous and Li (Theorem 2.18 in [5]):…”

“…However the converse of Theorem A in [5] has been specialized to commutators associated with the analytic reproducing kernel of a bounded and strictly pseudoconvex domain in C n . It is natural to ask whether the converse of Theorem A holds for the upper half-space setting (surely we need to impose additional restrictions on kernel k because (U) alone is not enough).…”

“…We also should point out that the compactness result for commutators in [5] can not be applied to the upper half-space setting right away. The hypotheses of Theorem 3.5 in [5] include that the homogeneous space X is compact, the kernel k(x, y) vanishes if either the n-th coordinate of x or that of y exceeds a certain number.…”

“…for all x, y ∈ H, then k is said to satisfy the estimate (U) (A kernel which satisfies (U) is said to be a 1-homogeneous kernel in [5]). Let M f and I k denote the multiplication operator with symbol f and the integral operator with kernel k defined by…”

The Boundary Behavior of Holomorphic Functions: Global and Local Results

“…Some relevant references would be [AHN], [[BEL1], [BEL2], [HOR], and [KRL1]. We defer the exploration of this more general situation to another time.…”