Heungsu Yi

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.

Representations and interpolations of harmonic Bergman functions on half-spaces

On the setting of the half-space of the euclidean n-space, we prove representation theorems and interpolation theorems for harmonic Bergman functions in a constructive way. We also consider the harmonic (little) Bloch spaces as limiting spaces. Our results show that well-known phenomena for holomorphic cases continue to hold. Our proofs of representation theorems also yield a uniqueness theorem for harmonic Bergman functions. As an application of interpolation theorems, we give a distance estimate to the harmonic little Bloch space. In the course of the proofs, pseudohyperbolic balls are used as substitutes for Bergman metric balls in the holomorphic case.

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.

Gleason's problem for harmonic Bergman and Bloch functions on half-spaces

On the setting of the half-spaceRn−1×R+, we investigate Gleasons problem for harmonic Bergman and Bloch functions. We prove that Gleasons problem for the harmonicLp-Bergman space is solvable if and only ifp>n. We also prove that Gleasons problem for the harmonic (little) Bloch space is solvable.

HARMONIC CONJUGATES OF WEIGHTED HARMONIC BERGMAN FUNCTIONS ON HALF-SPACES

On the setting of the upper half-space of the Euclidean space , we show that to each weighted harmonic Bergman function , there corresponds a unique conjugate system (,…, ) of u satisfying with an appropriate norm bound.

Bergman norm estimates of Poisson integrals

Abstract. On the half space Rn × R+ , it has been known that harmonic Bergman space bp can contain a positive function only if p > 1 + 1 n . Thus, for 1 ≤ p ≤ 1 + 1 n , Poisson integrals can be bp-functions only by means of their boundary cancellation properties. In this paper, we describe what those cancellation properties explicitly are. Also, given such cancellation properties, we obtain weighted norm inequalities for Poisson integrals. As a consequence, under weighted integrability condition given by our weighted norm inequalities, we show that our cancellation properties are equivalent to the bp-containment of Poisson integrals for p under consideration. Our results are sharp in the sense that orders of our weights cannot be improved.

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.