• East Asian mathematical journal
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• v.23 no.1
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• pp.37-43
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• 2007

We study a maximal operator defined on spaces of homogeneous type, and we prove that this operator is of weak type (1,1). As a consequence we show that the maximal operator belongs to the Muckenhoupt's class $A_1$.

• Bulletin of the Korean Mathematical Society
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• v.55 no.6
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• pp.1791-1809
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• 2018

Let T be the singular integral operator with nonsmooth kernel which was introduced by Duong and McIntosh, and $T_q(q{\in}(1,{\infty}))$ be the vector-valued operator defined by $T_qf(x)=({\sum}_{k=1}^{\infty}{\mid}T\;f_k(x){\mid}^q)^{1/q}$. In this paper, by proving certain weak type endpoint estimate of L log L type for the grand maximal operator of T, the author establishes some quantitative weighted bounds for $T_q$ and the corresponding vector-valued maximal singular integral operator.

• Journal of the Chungcheong Mathematical Society
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• v.8 no.1
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• pp.29-36
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• 1995

• Journal of the Korean Mathematical Society
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• v.34 no.2
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• pp.371-381
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• 1997

Our basic concern in this paper is to investigate some geometric properties of the graph of a maximal monotone operator in the one dimensional case. Using a well-known theorem of Minty, we answer S. Simon's questions affirmatively in the one dimensional case. Further developments of these results are also treated. In addition, we provide a new proof of Rockafellar's characterization of maximal monotone operators on R: every maximal monotne operator from R to $2^R$ is the subdifferential of a proper convex lower semicontinuous function.

• Journal of the Korean Mathematical Society
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• v.49 no.2
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• pp.395-404
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• 2012

Our first aim of this paper is to define maximal operators a-quadratic variation and of a-conditional quadratic variation for vectorvalued tree martingales and to show that these maximal operators and maximal operators of vector-valued tree martingale transforms are all sublinear operators. The second purpose is to prove that maximal operator inequalities of a-quadratic variation and of a-conditional quadratic variation for vector-valued tree martingales hold provided 2 ${\leq}$ a < $\infty$ by means of Marcinkiewicz interpolation theorem. Based on a result of reference [10] and using Marcinkiewicz interpolation theorem, we also propose a simple proof of maximal operator inequalities for vector-valued tree martingale transforms, under which the vector-valued space is a UMD space.

• Journal of the Korean Mathematical Society
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• v.35 no.2
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• pp.361-370
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• 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
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• v.23 no.2
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• pp.207-213
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• 2010

In this paper, we define the q-extension of the Hardy-Littlewood-type maximal operator related to q-Volkenborn integral. By the meaning of the extension of q-Volkenborn integral, we obtain the boundedness of the q-extension of the Hardy-Littlewood-type maximal operator in the p-adic integer ring.

• Bulletin of the Korean Mathematical Society
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• v.38 no.1
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• pp.93-100
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• 2001

Author proves weak type estimates of the maximal function associated with the Bochner-Riesz means while it is claimed p=2n/(n+1+$2\delta) and 0<\delta\leq(n-1)/2$ that the maximal function is bounded on L^p-{rad}$.

• Bulletin of the Korean Mathematical Society
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• v.61 no.1
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• pp.207-216
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• 2024

This note is devoted to establishing the variation continuity of the one-dimensional discrete uncentered multilinear maximal operator. The above result is based on some refine variation estimates of the above maximal functions on monotone intervals. The main result essentially improves some known ones.

• Journal of the Korean Mathematical Society
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• v.59 no.3
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• pp.495-517
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• 2022

Let M_α be a bilinear fractional maximal operator and BM_α be a fractional maximal operator associated with the bilinear Hilbert transform. In this paper, the compactness on weighted Lebesgue spaces are considered for commutators of bilinear fractional maximal operators; these commutators include the fractional maximal linear commutators M^j_α,β and BM^j_α,β (j = 1, 2), the fractional maximal iterated commutator ${\mathcal{M}}_{{\alpha},{\vec{b}}}$, and $BM_{{\alpha},{\vec{b}}}$, where b ∈ BMO(ℝ^d) and ${\vec{b}}\;=\;(b_1,b_2)\;{\in}\;BMO({\mathbb{R}}^d)\;{\times}\;BMO({\mathbb{R}}^d)$. In particular, we improve the well-known results to a larger scale for 1/2 < q < ∞ and give positive answers to the questions in [2].