In this paper, we investigate the weighted theory of Riesz transforms \(R_{{\Delta _N}}\) associated with the Neumann Laplacian and its commutator \(C_b(R_{{\Delta _N}})\) . The main goal is to show weighted compactness of \(C_b(R_{{\Delta _N}})\) on \(L^p(w)\) for any \(p \in (1, \infty )\) , \(w \in A_{p, {\Delta _N}}\) , and \(b \in \operatorname {CMO}_{{\Delta _N}}(\mathbb {R}^n)\) . To achieve this, we establish sharp weighted estimates, weighted endpoint estimates, local decay estimates for \(R_{{\Delta _N}}\) and \(C_b(R_{{\Delta _N}})\) . Moreover, we introduce the Hardy-Littlewood maximal operator associated with the Neumann Laplacian in order to characterize \(A_{p, {\Delta _N}}\) weights class. Beyond that, we give Rubio de Francia extrapolation theorems to prove weighted compactness of \(C_b(R_{{\Delta _N}})\) .
