• Communications of the Korean Mathematical Society
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• v.25 no.1
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• pp.27-36
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• 2010

We give a simple proof of certain mapping properties of the p-adic Riesz potential and Bessel potential, and the p-adic version of the Sobolev embedding theorem obtained in [6].

• Bulletin of the Korean Mathematical Society
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• v.58 no.1
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• pp.235-251
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• 2021

We consider the Schrödinger type operator ��k = (-∆)k+Vk on ℝn(n ≥ 2k + 1), where k = 1, 2 and the nonnegative potential V belongs to the reverse Hölder class RHs with n/2 < s < n. In this paper, we establish the (Lp, Lq)-boundedness of the higher order Riesz transform T��,�� = V2��∇2��-��2 (0 ≤ �� ≤ 1/2 < �� ≤ 1, �� - �� ≥ 1/2) and its adjoint operator T∗��,�� respectively. We show that T��,�� is bounded from Hardy type space $H^1_{\mathcal{L}_2}({\mathbb{R}}_n)$ into Lp2 (ℝn) and T∗��,�� is bounded from ��p1 (ℝn) into BMO type space $BMO_{\mathcal{L}_1}$ (ℝn) when �� - �� > 1/2, where $p_1={\frac{n}{4({\beta}-{\alpha})-2}}$, $p_2={\frac{n}{n-4({\beta}-{\alpha})+2}}$. Moreover, we prove that T��,�� is bounded from $BMO_{\mathcal{L}_1}({\mathbb{R}}_n)$ to itself when �� - �� = 1/2.

• Bulletin of the Korean Mathematical Society
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• v.59 no.5
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• pp.1255-1268
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• 2022

We consider the Schrödinger type operator 𝓛 = (-𝚫_ℍⁿ)² + V² on the Heisenberg group ℍⁿ, where 𝚫_ℍⁿ is the sub-Laplacian and the non-negative potential V belongs to the reverse Hölder class RH_s for s ≥ Q/2 and Q ≥ 6. We shall establish the (L^p, L^q) estimates for the Riesz transforms T_𝛼,𝛽,j = V^2𝛼𝛁^j_ℍⁿ𝓛^-𝛽, j = 0, 1, 2, 3, where 𝛁_ℍⁿ is the gradient operator on ℍⁿ, 0 < α ≤ 1-j/4, j/4 < 𝛽 ≤ 1, and 𝛽 - 𝛼 ≥ j/4.

• Bulletin of the Korean Mathematical Society
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• v.56 no.5
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• pp.1117-1127
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• 2019

Let ${\mathcal{L}}_2=(-{\Delta})^2+V^2$ be the $Schr{\ddot{o}}dinger$ type operator, where nonnegative potential V belongs to the reverse $H{\ddot{o}}lder$ class $RH_s$, s > n/2. In this paper, we consider the operator $T_{{\alpha},{\beta}}=V^{2{\alpha}}{\mathcal{L}}^{-{\beta}}_2$ and its conjugate $T^*_{{\alpha},{\beta}}$, where $0<{\alpha}{\leq}{\beta}{\leq}1$. We establish the $(L^p,\;L^q)$-boundedness of operator $T_{{\alpha},{\beta}}$ and $T^*_{{\alpha},{\beta}}$, respectively, we also show that $T_{{\alpha},{\beta}}$ is bounded from Hardy type space $H^1_{L_2}({\mathbb{R}}^n)$ into $L^{p_2}({\mathbb{R}}^n)$ and $T^*_{{\alpha},{\beta}}$ is bounded from $L^{p_1}({\mathbb{R}}^n)$ into BMO type space $BMO_{{\mathcal{L}}1}({\mathbb{R}}^n)$, where $p_1={\frac{n}{4({\beta}-{\alpha})}}$, $p_2={\frac{n}{n-4({\beta}-{\alpha})}}$.

• 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).

• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.423-435
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• 2014

In this paper, the fractional Hardy-type operator of variable order ${\beta}(x)$ is shown to be bounded from the Herz-Morrey spaces $M\dot{K}^{{\alpha},{\lambda}}_{p_1,q_1({\cdot})}(\mathbb{R}^n)$ with variable exponent $q_1(x)$ into the weighted space $M\dot{K}^{{\alpha},{\lambda}}_{p_2,q_2({\cdot})}(\mathbb{R}^n,{\omega})$, where ${\omega}=(1+|x|)^{-{\gamma}(x)}$ with some ${\gamma}(x)$ > 0 and $1/q_1(x)-1/q_2(x)={\beta}(x)/n$ when $q_1(x)$ is not necessarily constant at infinity. It is assumed that the exponent $q_1(x)$ satisfies the logarithmic continuity condition both locally and at infinity that 1 < $q_1({\infty}){\leq}q_1(x){\leq}(q_1)+$ < ${\infty}(x{\in}\mathbb{R}^n)$.

• Journal of the Korean Mathematical Society
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• v.58 no.6
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• pp.1327-1345
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• 2021

In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation of Chern-Simons-Higgs type $$u(x)=\vec{\;l\;}+C_{\ast}{{\displaystyle\smashmargin{2}{\int\nolimits_{\mathbb{R}^n}}}\;{\frac{(1-{\mid}u(y){\mid}^2){\mid}u(y){\mid}^2u(y)-\frac{1}{2}(1-{\mid}u(y){\mid}^2)^2u(y)}{{\mid}x-y{\mid}^{n-{\alpha}}}}dy.$$ Here u : ℝⁿ → ℝ^k is a bounded, uniformly continuous function with k ⩾ 1 and 0 < α < n, $\vec{\;l\;}{\in}\mathbb{R}^k$ is a constant vector, and C_* is a real constant. We prove that ${\mid}\vec{\;l\;}{\mid}{\in}\{0,\frac{\sqrt{3}}{3},1\}$ if u is the finite energy solution. Further, if u is also a differentiable solution, then we give a Liouville type theorem, that is either $u{\rightarrow}\vec{\;l\;}$ with ${\mid}\vec{\;l\;}{\mid}=\frac{\sqrt{3}}{3}$, when |x| → ∞, or $u{\equiv}\vec{\;l\;}$, where ${\mid}\vec{\;l\;}{\mid}{\in}\{0,1\}$.