Establish an appropriate Moser–Trudinger inequality for the unresolved extremal-region estimate

Establish an appropriate Moser–Trudinger-type inequality sufficient to justify the estimate of the integral over the region \(\{x\in\mathbb{R}^n:F^0(x)\leq R_\varepsilon,\ |v_k|\geq T_\varepsilon\}\) under the assumptions \(0<\lambda\leq\lambda_n\), \(\|F(\nabla v_k)\|_n\leq1\), and \(\|v_k\|_n=1\), thereby resolving the gap in the cited proof of the anisotropic exact-growth extremal result for the case \(\beta=0\).

Background

The paper identifies a gap in an earlier proof concerning the special case β=0\beta=0 of the anisotropic Moser–Trudinger inequality with exact growth. The disputed step attempts to control the contribution from the region where the Moser-type sequence is both spatially localized and has large amplitude, using only the pointwise estimate (1+b∣vk∣n/(n−1))−1≤ε(1+b|v_k|^{n/(n-1)})^{-1}\leq\varepsilon.

According to the remark, this argument requires an appropriate Moser–Trudinger-type inequality that controls the exponential integral on the specified region uniformly for sequences satisfying the gradient and LnL^n-normalization constraints. The authors state that such an inequality does not appear to be available in the existing literature and therefore revisit the case separately with a corrected argument.

References

Indeed, to justify the estimate

\int_{{x\in\mathbb{R}n\colon\, F0(x)\leq R_\varepsilon,\, |v_k|\geq T_\varepsilon}\left( \frac{\Phi(\lambda|v_k|{\frac{n}{n-1}})}{1+b|v_k|{\frac{n}{n-1}}}-\frac{\lambda{n-1}}{(n-1)!}|v_k|n \right)\,\mathrm{d}x\leq\varepsilon\int_{\mathbb{R}n}\Phi(\lambda|v_k|{\frac{n}{n-1}})\,\mathrm{d}x\leq\varepsilon C,

one would require an appropriate Moser--Trudinger type inequality, which does not seem to be available in the existing literature.

— Best Constants for Anisotropic Moser-Trudinger Inequalities with the Exact Growth Condition and Their Applications  (2610.12260 - Figueiredo et al., 8 Oct 2026) in Remark following Lemma 7.2, Section 7