Maximal regularity of parabolic equations associated with a discrete Laplacian

Abstract

Let Δd be the discrete Laplacian defined on Zd by settingΔdf(n→)=∑j=1d−[f(n→+e→j)+f(n→−e→j)−2f(n→)],n→∈Zd, where {e→j:j=1,…,d} is the standard basis for Rd.In this paper, we prove weighted mixed norm estimates and end-point estimates for the maximal regularity of the discrete parabolic equation{ut+Δdu=f,t∈[0,T)u(0,⋅)=0, where T∈(0,∞).