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 space \(H^{0,1}(\mathbb {R}^n)\) 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 [18], as illustrated in the simulations.
