• Journal of the Korean Mathematical Society
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• v.60 no.1
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• pp.33-69
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• 2023

Let α ∈ (0, ∞), p ∈ (0, ∞) and q(·) : ℝⁿ → [1, ∞) satisfy the globally log-Hölder continuity condition. We introduce the weak Herz-type Hardy spaces with variable exponents via the radial grand maximal operator and to give its maximal characterizations, we establish a version of the boundedness of the Hardy-Littlewood maximal operator M and the Fefferman-Stein vector-valued inequality on the weak Herz spaces with variable exponents. We also obtain the atomic and the molecular decompositions of the weak Herz-type Hardy spaces with variable exponents. As an application of the atomic decomposition we provide various equivalent characterizations of our spaces by means of the Lusin area function, the Littlewood-Paley g-function and the Littlewood-Paley $g^*_{\lambda}$-function.

• Journal of the Chungcheong Mathematical Society
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• v.36 no.2
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• pp.87-91
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• 2023

It is concerned with the size of bounded functions in Besov spaces. It reports that every closed subspace consisted with bounded functions in Besov spaces B^s_p,p(𝕋^d) (s < 0) is finite dimensional.

• 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.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.1
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• pp.117-121
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• 2009

For $1{\leq}p$, $q{\leq}{\infty}$ and $s{\in}\mathbb{R}$, it is proved that there exists a constant C > 0 such that for any $f{\in}B^{s+2}_{p,q}(\mathbb{R}^n)$ $${\parallel}f{\parallel}_{B^{s+2}_{p,q}(\mathbb{R}^n)}{\leq}C{\parallel}f\;-\;{\Delta}f{\parallel}_{B^{s}_{p,q}(\mathbb{R}^n)}$$, which tells us that the operator $I-\Delta$ is $B^{s+2}_{p,q}$-coercive on the Besov space $B^s_{p,q}$.

• Bulletin of the Korean Mathematical Society
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• v.55 no.1
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• pp.267-296
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• 2018

This paper studies the Cauchy problem and blow-up phenomena for a new generalized Camassa-Holm equation with cubic nonlinearity in the nonhomogeneous Besov spaces. First, by means of the Littlewood-Paley decomposition theory, we investigate the local well-posedness of the equation in $B^s_{p,r}$ with s > $max\{{\frac{1}{p}},\;{\frac{1}{2}},\;1-{\frac{1}{p}}\},\;p,\;r{\in}[0,{\infty}]$. Second, we prove that the equation is locally well-posed in $B^s_{2,r}$ with the critical index $s={\frac{1}{2}}$ by virtue of the logarithmic interpolation inequality and the Osgood's Lemma, and it is shown that the data-to-solution mapping is $H{\ddot{o}}lder$ continuous. Finally, we derive two kinds of blow-up criteria for the strong solution by using induction and the conservative property of m along the characteristics.