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Truong Dinh Dat

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Preprint Jul 2026

Stability and strong convergence for complex Hessian equations with $L^1$ data

We study complex $m$-Hessian equations on bounded hyperconvex domains with right-hand side in $L^1(\Omega)$. The main contribution of this paper is a strong stability result for weak solutions in the Hessian energy sense. More precisely, if $f_j \to f$ in $L^1(\Omega)$ and $u_j, u$ are the corresponding solutions, then \[ \int_\Omega |u_j-u|\, H_m(u_j) \longrightarrow 0. \] This provides convergence in the natural energy topology associated to the Hessian operator, which is significantly stronger than convergence in capacity. For completeness, we also recall the existence of solutions and stability in capacity, which follow from known results in the literature.

Truong Dinh Dat · 0 citations

Preprint Jul 2026

Uniform $L^\infty$ estimates for complex hessian equations on compact Hermitian manifolds

We develop a pluripotential approach to complex Hessian equations on compact Hermitian manifolds. In this setting, the lack of closedness of the background metric introduces torsion terms that prevent a direct extension of the K\"ahler theory. Our main result is a uniform $L^\infty$ estimate for bounded $\omega$-$m$-subharmonic solutions of the equation \[ (\omega + dd^c u)^m \wedge \omega^{n-m} = cf\,\omega^n, \] under the assumption that $f \in L^p$, $f \ge 0$ for some $p>1$. The proof combines a weak comparison principle with torsion error, a capacity theory adapted to the Hermitian setting, and a nonlinear iteration scheme controlling the decay of sublevel sets. As applications, we obtain existence, stability and compactness results for weak solutions with $L^p$ densities. These results extend several aspects of the pluripotential theory. of complex Hessian equations beyond the K\"ahler framework.

Truong Dinh Dat · 0 citations