Hyungwoon Koo

Commuting Toeplitz operators on the polydisk

We obtain characterizations of (essentially) commuting Toeplitz operators with pluriharmonic symbols on the Bergman space of the polydisk. We show that commuting and essential commuting properties are the same for dimensions bigger than 2, while they are not for dimensions less than or equal to 2. Also, the corresponding results for semi-commutators are obtained.

Positive Toeplitz operators between the harmonic Bergman spaces

On the setting of the upper half space we study positive Toeplitz operators between the harmonic Bergman spaces. We give characterizations of bounded and compact positive Toeplitz operators taking a harmonic Bergman space bp into another bq for 1<p<∞, 1<q<∞. The case p=1 or q=1 seems more intriguing and is left open for further investigation. Also, we give criteria for positive Toeplitz operators acting on b2 to be in the Schatten classes. Some applications are also included.

Sums of Toeplitz products with harmonic symbols

On the Bergman space of the unit disk, we consider a class of operators which contain sums of finitely many Toeplitz products with harmonic symbols. We give characterizations of when an operator in that class has finite rank or is compact. Our results provide a unified way of treating several known results.

Positive Toeplitz Operators of Schatten–Herz Type

Motivated by a recent work of Loaiza et al. for the holomorphic case on the disk, we introduce and study the notion of Schatten-Herz type Toeplitz operators acting on the harmonic Bergman space of the ball. We obtain characterizations of positive Toeplitz operators of Schatten-Herz type in terms of averaging functions and Berezin transforms of symbol functions. Our characterization in terms of Berezin transforms settles a question posed by Loaizaet al.

Composition operators acting on holomorphic Sobolev spaces

We study the action of composition operators on Sobolev spaces of analytic functions having fractional derivatives in some weighted Bergman space or Hardy space on the unit disk. Criteria for when such operators are bounded or compact are given. In particular, we find the precise range of orders of fractional derivatives for which all composition operators are bounded on such spaces. Sharp results about boundedness and compactness of a composition operator are also given when the inducing map is polygonal.

WEIGHTED HARMONIC BERGMAN FUNCTIONS ON HALF-SPACES

On the setting of the upper half-space H of the Eu- clidean n-space, we show the boundedness of weighted Bergman projection for 1 < p < 1 and nonorthogonal projections for 1 • p < 1. Using these results, we show that Bergman norm is equiva- lent to the normal derivative norms on weighted harmonic Bergman spaces. Finally, we flnd the dual of b 1.

Two-weighted inequalities for the derivatives of holomorphic functions and Carleson measures on the unit ball

We characterize those positive measure µ ’s on the higher dimensional unit ball such that “two-weighted inequalities” hold for holomorphic functions and their derivatives. Characterizations are given in terms of the Carleson measure conditions. The results of this paper also distinguish between the fractional and the tangential derivatives.

Revisit to a Theorem of Wogen

In this note we provide a new proof of a theorem of Wogen on the boundedness criterion for composition operators on Hardy space H2(Un) induced by holomorphic self-maps of the unit ball Un, and then generalize it to more general inducing self-maps.

Carleson measures for Bergman spaces and their dual Berezin transforms

We introduce the notion of weighted dual Berezin transforms and characterize Carleson measures for weighted Bergman spaces over the ball by a certain BMO property of their dual Berezin transforms.

Harmonic Bergman functions as radial derivatives of Bergman functions

In the setting of the half-space of the euclidean n-space, we show that every harmonic Bergman function is the radial derivative of a Bergman function with an appropriate norm bound.