Prove
Lemma 39 ([Continuous Sobolev functions in the closed form]). If u∈C(S^2)∩W^…Theorem 7.1 ([From energies and Green bounds to path laws]). In the…Proposition
Put u_n=R^n_α(fπ_n) . The maximum principle and the resolvent equation give \|u_n\|_∞≤(\|f\|_∞)/(α), H_nu_n=μ_n g_n, g_n=2(fπ_n-αu_n). In particular g_n is uniformly bounded, and multiplying the equation by u_n shows that _n(u_n) is uniformly bounded. We first prove equicontinuity. Fix a point x∈S^2 , and choose disjoint open neighborhoods U of x and V of a different point. Take μ -null boundaries, with μ(V)>0 . The neighborhood U may be made arbitrarily small in μ -mass because μ has no atoms. Write U_n=π_n^ -1 (U) and V_n'=π_n^ -1 (V) , and set q_n=g_n\1_ U_n - μ_n(g_n\1_ U_n ) μ_n(V_n') \1_ V_n' . For large n , the denominator is bounded below. Hence q_n has mean zero, bounded supremum, and μ_n(|q_n|)≤C_1μ_n(U_n) . Scaling yields \|G_nq_n\|_∞≤C_2μ_n(U_n)^ρ. Also _n(G_nq_n)=μ_n((G_nq_n)q_n) ≤\|G_nq_n\|_∞μ_n(|q_n|). Thus u_n-G_nq_n has bounded energy and supremum and is harmonic on U_n . Hypothesis~(iii) makes this difference equicontinuous on compact subsets of U . Given an oscillation tolerance, first choose U so that G_nq_n has smaller supremum, and then use harmonic equicontinuity inside U . A finite cover of S^2 proves equicontinuity of the original sequence. Boundedness and density of π_n(V_n) now give uniformly convergent subsequences. Let u be one such limit. Hypothesis~(ii) and Lemma 39 ([Continuous Sobolev functions in the closed form]) show that u∈ F∩C(S^2) . The discrete resolvent minimizes J_n(v)=12_n(v)+αμ_n(v^2) -2μ_n((fπ_n)v). The energy lower bound, uniform convergence, and weak convergence of the measures imply (u,u)+αμ(u^2)-2μ(fu) ≤_n J_n(u_n). For any smooth v , take its uniform recovery sequence v_n . Minimality gives J_n(u_n)≤J_n(v_n) , and the latter converges to (v,v)+αμ(v^2)-2μ(fv) . Smooth functions are a core of the closed form by its definition, so u minimizes this functional over all of F . Strict convexity for α>0 identifies u=R_αf in L^2(μ) . Full support makes its continuous version unique. Every subsequence has the same limit, which proves. Finally, \|R^n_αa-R^n_αb\|_∞ ≤α^ -1 \|a-b\|_∞ . This contraction bound proves the claim for varying right-hand sides.