• Communications of the Korean Mathematical Society
• /
• v.37 no.1
• /
• pp.91-103
• /
• 2022

In this paper, we introduce Toeplitz operators and Hankel operators with complex Borel measures on the closed unit disk. When a positive measure 𝜇 on (-1, 1) is a Carleson measure, it is known that the corresponding Hankel matrix is bounded and vice versa. We show that for a positive measure 𝜇 on 𝔻, 𝜇 is a Carleson measure if and only if the Toeplitz operator with symbol 𝜇 is a densely defined bounded linear operator. We also study Hankel operators of Hilbert-Schmidt class.

• Journal of the Korean Mathematical Society
• /
• v.32 no.4
• /
• pp.649-659
• /
• 1995

The purpose of this paper is to generalize the Carleson inequality, which is known to play important roles in harmonic analysis. The result given here is a generalization of Coifmann, Meyer, Stein [CMS]. A similar result is shown by Deng [D].

• Journal of the Korean Mathematical Society
• /
• v.35 no.1
• /
• pp.11-21
• /
• 1998

A condition for a certain maximal operator to be of strong type (p,p) is characterized in terms of Carleson measure.

• Communications of the Korean Mathematical Society
• /
• v.17 no.1
• /
• pp.31-36
• /
• 2002

In this paper we first introduce a space of homogeneous type X, and we consider a kind of generalized upper half-space X $\times$ (0, $\infty$). We are mainly concerned with some inequalities in terms of Carleson measures or in terms of certain maximal operators with respect to general approach regions in X $\times$ (0, $\infty$). The main tool of the proof is the Whitney decomposition.

• Bulletin of the Korean Mathematical Society
• /
• v.53 no.3
• /
• pp.711-722
• /
• 2016

We study Toeplitz operators $T_{\nu}$ on generalized Fock spaces $F^2_{\phi}$ with a locally finite positive Borel measures ${\nu}$ as symbols. We characterize operator-theoretic properties (boundedness and compactness) of $T_{\nu}$ in terms of the Fock-Carleson measure and the Berezin transform ${\tilde{\nu}}$.

• Journal of the Chungcheong Mathematical Society
• /
• v.20 no.3
• /
• pp.333-342
• /
• 2007

Suppose that ${\mu}$ is a finite positive Borel measure on the unit ball $B{\subset}C^n$. The boundary of B is the unit sphere $S=\{z:{\mid}z{\mid}=1\}$. Let ${\sigma}$ be the rotation-invariant measure on S such that ${\sigma}(S)=1$. In this paper, we will show that if $sup_{{\zeta}{\in}S}\;{\int}_{B}\;P(z,{\zeta})d{\mu}(z)$<${\infty}$ where $P(z,{\zeta})$ is the Poission-Szeg$\ddot{o}$ kernel for B, then ${\mu}$ is a Carleson measure. We will also show that if $sup_{{\zeta}{\in}S}\;{\int}_{B}\;P(z,{\zeta})d{\mu}(z)$<${\infty}$, then the operator T such that T(f) = P[f] is compact as a mapping from $L^p(\sigma)$ into $L^p(B,d{\mu})$.

• Bulletin of the Korean Mathematical Society
• /
• v.33 no.4
• /
• pp.497-501
• /
• 1996

Let B denote the unit ball in $C^n(n \geq 1)$, and $\upsilon$ the 2n-dimensional Lebesgue measure on B normalized so that $\upsilon(B) = 1$, while $\sigma$ is the normalized surface measure on the boundary S of B.

• Journal of the Korean Mathematical Society
• /
• v.35 no.2
• /
• pp.361-370
• /
• 1998

A condition for a certain maximal operator to be of weak type (p, p) is studied. This operator unifies various maximal operators cited in the literatures.

• Journal of the Chungcheong Mathematical Society
• /
• v.10 no.1
• /
• pp.141-146
• /
• 1997

In this paper we prove that the corona theorem for the algebra $H^{\infty}(D){\cap}D(D)$. That is, we prove that $\mathcal{M}{\setminus}{\overline{D}}$ is an empty set where $\mathcal{M}$ is the maximal ideal space of the given algebra.

• Bulletin of the Korean Mathematical Society
• /
• v.35 no.2
• /
• pp.293-299
• /
• 1998

In this note we consider some characterizations of analytic functions of bounded and vanishing mean oscillation on the unit disk in $\mathbb{C}$ and answer a question about them in the negative.