Kazhdan-Lusztig cells in some weighted Coxeter groups

Let (W, S) be a Coxeter group with S = I ⊔J such that J consists of all universal elements of S and that I generates a finite parabolic subgroup WI of W with w0 the longest element of WI. We describe all the left cells and two-sided cells of the weighted Coxeter group (W,S,L) that have non-empty intersection with WJ, where the weight function L of (W,S) is in one of the following cases: (i) max{L(s) | s ∈ J} < min{L(t) | t ∈ I}; (ii) min{L(s) | s ∈ J} ⩾ L(w0); (iii) there exists some t ∈ I satisfying L(t) < L(s) for any s ∈ I − {t} and L takes a constant value LJ on J with LJ in some subintervals of [1,L(w0) − 1]. The results in the case (iii) are obtained under a certain assumption on (W,WI).