Fix $\varepsilon>0$ and $\alpha>0$ to be chosen later. It follows from (2) and propositions 2.5, 3.1 that

$$||\varphi-\psi||_{L^{\infty}(X)}\leq\varepsilon+C_{1}\left[Cap_{\omega}(|\varphi-\psi|>\varepsilon)\right]^{\alpha/n}.$$

Applying the refined version of lemma 2.2 which involves the uniform bound on $||\varphi||_{L^{\infty}(X)},||\psi||_{L^{\infty}(X)}$ (see inequality (3)), we obtain

$$Cap_{\omega}(|\varphi-\psi|>\varepsilon)\leq\frac{C_{2}}{\varepsilon^{n+2/q}}\int_{X}|\varphi-\psi|^{2/q}(f+g)\omega^{n}.$$

It follows thus from Hölder’s inequality that

$$Cap_{\omega}(|\varphi-\psi|>\varepsilon)\leq\frac{C_{3}||f+g||_{L^{p}}}{\varepsilon^{n+2/q}}\left[||\varphi-\psi||_{L^{2}(\omega^{n})}\right]^{2/q}.$$

Choose now $\varepsilon:=||\varphi-\psi||_{L^{2}}^{\omega}$ where $0<\gamma<2/(2+nq)$. Then

$$Cap_{\omega}(|\varphi-\psi|>\varepsilon)\leq C_{4}\left[||\varphi-\psi||_{L^{2}}\right]^{2/q-\gamma(n+2/q)}.$$

We infer

$$||\varphi-\psi||_{L^{\infty}(X)}\leq||\varphi-\psi||_{L^{2}}^{\gamma}+C_{5}||\varphi-\psi||_{L^{2}}^{\gamma^{\prime}},\;\text{ where }\gamma^{\prime}=\frac{\alpha}{n}\left[2/q-\gamma(n+2/q)\right].$$

We finally choose $\alpha>0$ so large that $\gamma\leq\gamma^{\prime}$ and adjust the value of the constant $C$: this yields the desired estimate. ∎