• Journal of the Korean Mathematical Society
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• v.59 no.2
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• pp.235-254
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• 2022

Let ℍⁿ be the Heisenberg group and Q = 2n + 2 be its homogeneous dimension. Let 𝓛 = -∆_ℍⁿ + V be the Schrödinger operator on ℍⁿ, where ∆_ℍⁿ is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class $B_{q_1}$ for q₁ ≥ Q/2. Let H^p_𝓛(ℍⁿ) be the Hardy space associated with the Schrödinger operator 𝓛 for Q/(Q+𝛿₀) < p ≤ 1, where 𝛿₀ = min{1, 2 - Q/q₁}. In this paper, we consider the Hardy type estimates for the operator T_𝛼 = V^𝛼(-∆_ℍⁿ + V )^-𝛼, and the commutator [b, T_𝛼], where 0 < 𝛼 < Q/2. We prove that T_𝛼 is bounded from H^p_𝓛(ℍⁿ) into L^p(ℍⁿ). Suppose that b ∈ BMO^𝜃_𝓛(ℍⁿ), which is larger than BMO(ℍⁿ). We show that the commutator [b, T_𝛼] is bounded from H¹_𝓛(ℍⁿ) into weak L¹(ℍⁿ).

• Communications of the Korean Mathematical Society
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• v.5 no.1
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• pp.49-53
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• 1990

• Communications of the Korean Mathematical Society
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• v.6 no.1
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• pp.97-99
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• 1991

• East Asian mathematical journal
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• v.11 no.2
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• pp.223-228
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• 1995

• East Asian mathematical journal
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• v.11 no.2
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• pp.175-185
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• 1995

• East Asian mathematical journal
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• v.9 no.1
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• pp.7-14
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• 1993

• Communications of the Korean Mathematical Society
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• v.2 no.1
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• pp.33-37
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• 1987

• Journal for History of Mathematics
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• v.27 no.4
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• pp.285-297
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• 2014

G. H. Hardy laid the foundation of classical studies on double Fourier series at the beginning of the 20th century. In this paper we are concerned not only with Fourier series but more generally with trigonometric series. We consider Norlund means and Cesaro summation method for double Fourier Series. In section 2, we investigate the classical results on the summability and the convergence of double Fourier series from G. H. Hardy to P. Sjolin in the mid-20th century. This study concerns with the $L^1(T^2)$-convergence of double Fourier series fundamentally. In conclusion, there are the features of the classical results by comparing and reinterpreting the theorems about double Fourier series mutually.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.12 no.4
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• pp.261-269
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• 2008

Let $B_1$ be a unit ball in $R^n(n{\geq}3)$, and $2^*=2n/(n-2)$ be the critical Sobolev exponent for the embedding $H_0^1(B_1){\hookrightarrow}L^{2^*}(B_1)$. By using a variant of Pohoz$\check{a}$aev's identity, we prove the nonexistence of nodal solutions for the Dirichlet problem $-{\Delta}u-{\mu}\frac{u}{{\mid}x{\mid}^2}={\lambda}u+{\mid}u{\mid}^{2^*-2}u$ in $B_1$, u=0 on ${\partial}B_1$ for suitable positive numbers ${\mu}$ and ${\nu}$.

• Interdisciplinary Bio Central
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• v.5 no.4
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• pp.8.1-8.3
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• 2013

Unlike the traditional view, it is not mysterious about how G. Mendel chose the seven characters of the pea, Pisum sativum, that he studied. He first chose the pea that met three conditions he set up and repeated experiments for two years. Apparently, he knew that those characters were controlled by countable elements. Then, he derived the prediction on the basis of his idea about the elements, and selected the seven characters that satisfied the prediction. He knew "no prediction no science". In population genetics the Hardy-Weinberg principle is well known and cited in many papers and books. However, Mendel already derived the same principle in his paper, because he was acquainted also with physics and mathematics. Actually, the principle was trivial when they derived, but not at all when Mendel did. It is also well known that Mendel's laws were forgotten and rediscovered at the term of the 19th century. That may not be true either. His laws were internationally well known before the rediscovery. In fact, the 1881-year version of the Encyclopedia Britannica contains his laws.