• 충청수학회지
• /
• 제5권1호
• /
• pp.159-165
• /
• 1992

In 1986, R. Huff [3] showed that a Dunford integrable function is Pettis integrable if and only if T : $X^*{\rightarrow}L_1(\mu)$ is weakly compact operator and {$T(K(F,\varepsilon))|F{\subset}X$, F : finite and ${\varepsilon}$ > 0} = {0}. In this paper, we introduce the notion of Bourgain property of real valued functions formulated by J. Bourgain [2]. We show that the class of pettis integrable functions is linear space and if lis bounded function with Bourgain property, then T : $X^{**}{\rightarrow}L_1(\mu)$ by $T(x^{**})=x^{**}f$ is $weak^*$ - to - weak linear operator. Also, if operator T : $L_1(\mu){\rightarrow}X^*$ with Bourgain property, then we show that f is Pettis representable.

• 대한수학회지
• /
• 제42권3호
• /
• pp.405-434
• /
• 2005

We establish appropriate $L^p$ estimates for a class of maximal operators $S_{\Omega}^{(\gamma)}$ on the product space $R^n\;\times\;R^m\;when\;\Omega$ lacks regularity and $1\;\le\;\gamma\;\le\;2.\;Also,\;when\;\gamma\;=\;2$, we prove the $L^p\;(2\;{\le}\;P\;<\;\infty)\;boundedness\;of\;S_{\Omega}^{(\gamma)}\;whenever\;\Omega$ is a function in a certain block space $B_q^{(0,0)}(S^{n-1}\;\times\;S^{m-1})$ (for some q > 1). Moreover, we show that the condition $\Omega\;{\in}\;B_q^{(0,0)}(S^{n-1}\;\times\;S^{m-1})$ is nearly optimal in the sense that the operator $S_{\Omega}^{(2)}$ may fail to be bounded on $L^2$ if the condition $\Omega\;{\in}\;B_q^{(0,0)}(S^{n-1}\;\times\;S^{m-1})$ is replaced by the weaker conditions $\Omega\;{\in}\;B_q^{(0,\varepsilon)}(S^{n-1}\;\times\;S^{m-1})\;for\;any\;-1\;<\;\varepsilon\;<\;0.$

• 대한수학회논문집
• /
• 제19권3호
• /
• pp.435-452
• /
• 2004

In this paper, we prove the weighted continuity of multilinear Marcinkiewicz operators for the extreme cases of p.

• 대한수학회지
• /
• 제37권4호
• /
• pp.613-624
• /
• 2000

We characterize a condition for M to be of weak type ($\Phi$1, $\Phi$2) in terms of Orlicz norms.

• 대한수학회보
• /
• 제48권1호
• /
• pp.141-149
• /
• 2011

We consider the problem to determine when a Toeplitz operator is bounded on weighted Bergman spaces. We show that Toeplitz operators induced by elements of some set are bounded and each element of the set is related with a Carleson measure on the weighted Bergman space.

• 통합자연과학논문집
• /
• 제4권4호
• /
• pp.289-293
• /
• 2011

In this paper we obtain some Sobolev estimates for the integral operator over singular curves (t, $t^m$) on $\mathbb{R}^2$ for $m{\geq}2$.

• 충청수학회지
• /
• 제33권1호
• /
• pp.57-63
• /
• 2020

We investigate the boundedness of the spectrum of a convex base function for a new function space. The result guarantees the continuity of the Calderón-Zygmund operators on the new space.

• Kyungpook Mathematical Journal
• /
• 제59권2호
• /
• pp.277-291
• /
• 2019

Using the concept of bounded boundary rotation, we investigate various properties of two new generalized classes of spirallike and Robertson functions of complex order with bounded boundary rotations.

• 대한수학회지
• /
• 제58권5호
• /
• pp.1299-1309
• /
• 2021

Demeter [1] and Kuk and Lee [5] proved the unboundedness of the trilinear Hilbert transforms H_a,b,c under the critical index 1/2 for some parameters a, b and c. We show the unboundedness of H_a,b,c for any parameters.

• 대한수학회논문집
• /
• 제37권1호
• /
• pp.75-89
• /
• 2022

In this paper, we prove new fixed point theorems for single-valued and multi-valued weakly Picard operators in complete metric spaces and give several examples. As applications, we give several results to Fredholm integral equation.