Validity of an alternative mollifier for ensuring convergence property (4.18)

Ascertain whether replacing the Gaussian mollifier δ_ϵ by the normalized indicator I_{|X|≤ϵ}/|I_{|X|≤ϵ}| in the approximation of the identity can be modified so that the convergence condition lim_{ϵ→ϵ0} [T^0_ϵ(Z1) − T^0_{ϵ0}(Z1)] I_{Ω×[0,T]}_∞ = 0 (Lemma 4.1, item (4)) still holds.

Background

To apply Leray–Schauder degree, the authors introduce approximating integral equations using a Gaussian mollifier δϵ and require a specific convergence property (4.18). They remark that choosing the normalized indicator I{|X|≤ϵ}/|I_{|X|≤ϵ}| instead of δ_ϵ would not satisfy (4.18) as stated, but speculate that with further refinement it might be made to work.

They explicitly note that they have not tested this alternative, leaving open whether a suitably modified indicator-based mollifier could fulfill the needed convergence criterion.