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

Abstract Let u_α(x,t), α ∈ (0,2) be the solution of the equation Δx,t u_α(x,t)+(1-α)t⁻¹∂ₜ u_α(x,t)=0 on Rⁿ⁺¹₊=Rⁿ×(0,∞) subject to u_α(x,0)=f(x) on Rⁿ. As the endpoint of the Poisson-Bessel potential u_α, the potential u₀(x,t) solves the equation Δx,t ((ln t⁻¹)u₀(x,t))+t⁻¹∂ₜ ((ln t⁻¹)u₀(x,t))=0 on Rⁿ⁺¹₊ subject to u₀(x,0)=f(x) on Rⁿ. The main goal of this paper is to characterize a nonnegative measure μ on Rⁿ⁺¹₊ such that f(x)↦ u_α(x,t) induces a bounded embedding from the fractional L¹-Hardy-Sobolev space Hα,1(Rⁿ), α ∈ (0,2) into the weak Lebesgue space WLqμ( Rⁿ⁺¹₊), q∈ [1,∞) and f(x)↦ u₀(x,t) induces a bounded embedding from the Hardy H0,1(Rⁿ) into the Lebesgue space Lqμ( Rⁿ⁺¹₊), q∈ [1,∞). Building upon the trace principles, we exploit Hα,1 space for image characterization instead of the bounded variation space. Our proposed (Hα,1, Lq) and (Hα,1, log) decomposition for image denoising demonstrate superior restorations, particularly in edges and texture preservation, when compared to the ROF model {ROF}, as illustrated in the simulations. 2020 Mathematics Subject Classification. 31C15, 42B35, 42B37, 28A78.