Trudinger-Moser type inequalities for the Hessian equation with logarithmic weights

TL;DR

This work develops sharp log-weighted Trudinger-Moser inequalities for the Hessian equation in $k$-admissible spaces, extending the classical Tian–Wang framework and complementing Calanchi–Ruf. A radial reduction to the one-dimensional weighted Sobolev space $X^{1,k+1}_{1,w}$ paired with Leckband’s integral inequality yields transported TM inequalities with explicit critical exponents $\gamma_{n,\beta}$ and thresholds $\alpha_{n,\beta}$; sharpness is established for both single and double exponential growth. The authors prove existence and regularity of log-weighted extremals, including attainability results for transported problems and the Hessian setting, thereby connecting transported extremals to genuine $k$-admissible maximizers in $\Phi^{k}_{0,\mathrm{rad}}(B,w)$. These results generalize prior work for the Laplacian and Hessian settings and provide a robust variational framework for critical exponential growth in fully nonlinear PDEs with logarithmic weights.

Abstract

We establish sharp Trudinger-Moser inequalities with logarithmic weights for the $k$-Hessian equation and investigate the existence of maximizers. Our analysis extends the classical results of Tian and Wang to $k$-admissible function spaces with logarithmic weights, providing a natural complement to the work of Calanchi and Ruf. Our approach relies on transforming the problem into a one-dimensional weighted Sobolev space, where we solve it using various techniques, including some radial lemmas and certain Hardy-type inequalities, which we establish in this paper, as well as a theorem due to Leckband.

Figures (1)

• Figure 1: Intuition for the choice of $F$

Theorems & Definitions (52)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.1
• proof : Proof of Corollary \ref{['thm0']}
• Proposition 2.1
• proof