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$L^p$ mapping properties for nonlocal Schrödinger operators with certain potentials

  • * Corresponding author: Yong-Cheol Kim

    * Corresponding author: Yong-Cheol Kim
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  • In this paper, we consider nonlocal Schrödinger equations with certain potentials $V∈{\rm{RH}}^q$($q>\frac{n}{2s}>1$ and $0<s <1$) of the form

    $\begin{equation*}L_K u+V u = f\,\,\text{ in }\; \mathbb{R}^n \end{equation*}$

    where $L_K$ is an integro-differential operator. We denote the solution of the above equation by $\mathcal{S}_V f: = u$, which is called the inverse of the nonlocal Schrödinger operator $L_K+V$ with potential $V$; that is, $\mathcal{S}_V = (L_K+V)^{-1}$. Then we obtain an improved version of the weak Harnack inequality of nonnegative weak subsolutions of the nonlocal equation

    $\begin{equation}\begin{cases}L_K u+V u = 0\,\,&\text{ in }\; \Omega,\\ u = g\,\,&\text{ in }\; \mathbb{R}^n\backslash\Omega, \;\;\;\;\;\;\;\;\; (1)\end{cases}\end{equation}$

    where $g∈ H^s(\mathbb{R}^n)$ and $\Omega$ is a bounded open domain in $\mathbb{R}^n$ with Lipschitz boundary, and also get an improved decay of a fundamental solution $\mathfrak{e}_V$ for $L_K+V$. Moreover, we obtain $L^p$ and $L^p-L^q$ mapping properties of the inverse $\mathcal{S}_V$ of the nonlocal Schrödinger operator $L_K+V$.

    Mathematics Subject Classification: Primary: 7G20, 45K05, 35J60, 35B65, 35D10; Secondary: 60J75.


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  • Figure 1.  The range of $(p,q)$ valid in Theorem 1.4

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