A trace principle for fractional Laplacian with an application to image process

Abstract Let $u_\alpha(x,t)$, $\alpha \in (0,2)$ be the solution of the equation $$\Delta_{x,t} u_\alpha(x,t)+(1-\alpha)t^{-1}\partial_t u_\alpha(x,t)=0$$ on $\mathbb R^{n+1}_+=\mathbb{R}^n\times(0,\infty)$ subject to $u_\alpha(x,0)=f(x)$ on $\mathbb{R}^n$. As the endpoint of the Poisson-Bessel potential $u_\alpha$, the potential $u_0(x,t)$ solves the equation $$ \Delta_{x,t} \big((\ln t^{-1})u_0(x,t)\big)+t^{-1}\partial_t \big((\ln t^{-1})u_0(x,t)\big)=0 $$ on $\mathbb R^{n+1}_+$ subject to $u_0(x,0)=f(x)$ on $\mathbb{R}^n$. The main goal of this paper is to characterize a nonnegative measure $\mu$ on $\mathbb R^{n+1}_+$ such that $f(x)\mapsto u_\alpha(x,t)$ induces a bounded embedding from the fractional $L^1$-Hardy-Sobolev space $H^{\alpha,1}(\mathbb{R}^n)$, $\alpha \in (0,2)$ into the weak Lebesgue space $WL^q_{\mu}(\mathbb R^{n+1}_+)$, $q\in [1,\infty)$ and $f(x)\mapsto u_0(x,t)$ induces a bounded embedding from the Hardy $H^{0,1}(\mathbb{R}^n)$ into the Lebesgue space $L^q_{\mu}(\mathbb R^{n+1}_+)$, $q\in [1,\infty)$. Building upon the trace principles, we exploit $H^{\alpha,1}$ space for image characterization instead of the bounded variation space. Our proposed $(H^{\alpha,1}, L^q)$ and $(H^{\alpha,1}, \log)$ decomposition for image denoising demonstrate superior restorations, particularly in edges and texture preservation, when compared to the ROF model \cite{ROF}, as illustrated in the simulations. 2020 Mathematics Subject Classification. 31C15, 42B35, 42B37, 28A78.

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