Lemma 35 ([Gap under doubling]). Fix κ∈(0,1/10) . The barriers in Lemma 34 ([Comparison barriers]) have a fixed constant γ>0 such that, whenever R₊=2r≤r_* , b_r-b_{R_}+ ≥γR₊^-4 R₊≤a≤(1+κ)R₊.

Lemma 34 ([Comparison barriers]). There are fixed positive constants α,ε₀,C'₀,c_b,c' with the following properties for all sufficiently small r_>0 . For 0<r≤r_ define Γ_r=ε_{0W_s}+C'_0r^4α W_s¹+α , B_r=W₀+q₀-Γ_r, b_r=B_r-V_c, …

Lemma 33 ([Kernel and potential estimates]). Uniformly as r_0 , a center contributing to Rf(y) satisfies |z-y|≤2t_y and t_z/t_y=1+O(r_^w) . Moreover, R1(y)=1+O(r_*^w), \|∇_z K_z(y)\|_{∞≤Ct_y}^-4 , \|∇_z K_z(y)\|_{2≤Ct_y}^-5/2 . The norms …

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The difference is exactly b_r-b_ R_+ =C'_0(1-2^ -4α )R_+^ 4α W_s^ 1+α . The lower bound for W_s gives the assertion with γ=C'_0(1-2^ -4α ) c_θ^ 1+α (1+κ)^ -4(1+α) >0.

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Lemma 35 ([Gap under doubling]). Fix κ∈(0,1/10) . The barriers in Lemma 34 ([Comparison barriers]) have a fixed constant γ>0 such that, whenever R+=2r≤r* , br-bR_+ ≥γR+-4 R+≤a≤(1+κ)R+.

Lemma 34 ([Comparison barriers]). There are fixed positive constants α,ε0,C'0,cb,c' with the following properties for all sufficiently small r*>0 . For 0<r≤r* define Γr=ε0Ws+C'0r4α Ws1+α , Br=W0+q0-Γr, br=Br-Vc, …

Lemma 33 ([Kernel and potential estimates]). Uniformly as r*0 , a center contributing to Rf(y) satisfies |z-y|≤2ty and tz/ty=1+O(r*w) . Moreover, R1(y)=1+O(r*w), \|∇z Kz(y)\|∞≤Cty-4 , \|∇z Kz(y)\|2≤Cty-5/2 . The norms …

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Lemma 35 ([Gap under doubling]). Fix κ∈(0,1/10) . The barriers in Lemma 34 ([Comparison barriers]) have a fixed constant γ>0 such that, whenever R₊=2r≤r_* , b_r-b_{R_}+ ≥γR₊^-4 R₊≤a≤(1+κ)R₊.

Lemma 34 ([Comparison barriers]). There are fixed positive constants α,ε₀,C'₀,c_b,c' with the following properties for all sufficiently small r_>0 . For 0<r≤r_ define Γ_r=ε_{0W_s}+C'_0r^4α W_s¹+α , B_r=W₀+q₀-Γ_r, b_r=B_r-V_c, …

Lemma 33 ([Kernel and potential estimates]). Uniformly as r_0 , a center contributing to Rf(y) satisfies |z-y|≤2t_y and t_z/t_y=1+O(r_^w) . Moreover, R1(y)=1+O(r_*^w), \|∇_z K_z(y)\|_{∞≤Ct_y}^-4 , \|∇_z K_z(y)\|_{2≤Ct_y}^-5/2 . The norms …