• Bulletin of the Korean Mathematical Society
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• v.35 no.3
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• pp.591-603
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• 1998

Taking into account the smoothness determined by sharp maximal functions, Sobolev type embedding results on Triebel-Lizorkin and Besov spaces are obtained.

• Bulletin of the Korean Mathematical Society
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• v.55 no.2
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• pp.659-672
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• 2018

In this note we prove that several classes of Littlewood-Paley square operators defined by the kernels without any regularity are bounded on Triebel-Lizorkin spaces $F^{p,q}_{\alpha}({\mathbb{R}}^n)$ and Besov spaces $B^{p,q}_{\alpha}({\mathbb{R}}^n)$ for 0 < ${\alpha}$ < 1 and 1 < p, q < ${\infty}$.

• Bulletin of the Korean Mathematical Society
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• v.61 no.2
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• pp.529-540
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• 2024

In this paper, using the Fourier transform, inverse Fourier transform and Littlewood-Paley decomposition technique, we prove the boundedness of bilinear pseudodifferential operators with symbols in the bilinear Hörmander class $BS^{m}_{1,1}$ in variable Triebel-Lizorkin spaces and variable Besov spaces.

• Bulletin of the Korean Mathematical Society
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• v.47 no.4
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• pp.839-857
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• 2010

We give a set of continuous characterizations for the homogeneous Triebel-Lizorkin spaces and use them to study boundedness properties of Fourier multiplier operators whose symbols satisfy a generalization of H$\ddot{o}$rmander's condition. As an application, we give new direct proofs of the imbedding theorems of the Sobolev type.

• Bulletin of the Korean Mathematical Society
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• v.59 no.6
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• pp.1539-1555
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• 2022

In this paper, we establish the boundedness and continuity for variation operators for θ-type Calderón-Zygmund singular integrals and their commutators on the Triebel-Lizorkin spaces. As applications, we obtain the corresponding results for the Hilbert transform, the Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators.

• Bulletin of the Korean Mathematical Society
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• v.58 no.2
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• pp.277-303
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• 2021

Under the rough kernels Ω belonging to the block spaces B0,qr (Sn-1) or the radial Grafakos-Stefanov kernels W����(Sn-1) for some r, �� > 1 and q ≤ 0, the boundedness and continuity were proved for two classes of rough maximal singular integrals and maximal operators associated to polynomial mappings on the Triebel-Lizorkin spaces and Besov spaces, complementing some recent boundedness and continuity results in [27, 28], in which the authors established the corresponding results under the conditions that the rough kernels belong to the function class L(log L)α(Sn-1) or the Grafakos-Stefanov class ����(Sn-1) for some α ∈ [0, 1] and �� ∈ (2, ∞).

• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1451-1469
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• 2013

We present various new sharp assertions on multipliers in mixed norm, weighted Hardy and new Lizorkin-Triebel spaces of harmonic functions in higher dimension. Some results are new even in onedimensional case.

• Journal of the Korean Mathematical Society
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• v.45 no.6
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• pp.1561-1576
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• 2008

In this paper we shall prove some weighted norm inequalities of the form $${\int}_{R^n}\;|Tf(x)|^pu(x)dx\;{\leq}\;C_p\;{\int}_{R^n}\;|f(x)|^pNu(x)dx$$ for certain rough singular integral T and maximal singular integral $T^*$. Here u is a nonnegative measurable function on $R^n$ and N denotes some maximal operator. As a consequence, some vector valued inequalities for both T and $T^*$ are obtained. We shall also get a boundedness result of T on the Triebel-Lizorkin spaces.

• Bulletin of the Korean Mathematical Society
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• v.56 no.5
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• pp.1099-1115
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• 2019

In the present paper, we establish certain $L^p$ bounds for the generalized parametric Marcinkiewicz integral operators associated to surfaces generated by polynomial compound mappings with rough kernels in Grafakos-Stefanov class ${\mathcal{F}}_{\beta}(S^{n-1})$. Our main results improve and generalize a result given by Al-Qassem, Cheng and Pan in 2012. As applications, the corresponding results for the generalized parametric Marcinkiewicz integral operators related to the Littlewood-Paley $g^*_{\lambda}$-functions and area integrals are also presented.