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Higher complex Sobolev spaces on complex manifolds

Thai Duong Do, Duc-Bao Nguyen

TL;DR

The work introduces higher complex Sobolev spaces $W^*_q$ on local domains and compact Kähler manifolds, equipping them with quasinorms and related capacities to study complex-analytic regularity. It proves Moser–Trudinger inequalities with exponents growing like $2^q$, establishes psh majorants to bound $|\varphi|^\alpha$, and derives both local and global exponential integrability results. The paper then connects these spaces to the complex Monge-Ampère operator, showing inclusions into energy classes $\mathcal E(X,\omega)$ and $\mathcal E^{q-1}(X,\omega)$ (global) and to the Cegrell–Blocki domain $\mathcal D(\Omega)$ (local), via inductive energy estimates. These results position $W^*_q$ as a robust framework that bridges Sobolev-type regularity, pluripotential theory, and Monge-Ampère equations, with implications for variational approaches and geometric analysis on complex manifolds. Overall, the work demonstrates that higher complex Sobolev regularity enhances Moser–Trudinger type control and aligns with established Monge-Ampère energy frameworks, enabling new tools for complex dynamics and Kähler geometry.

Abstract

We study higher complex Sobolev spaces and their corresponding functional capacities. In particular, we prove the Moser-Trudinger inequality for these spaces and discuss some relationships between these spaces and the complex Monge-Ampère equation.

Higher complex Sobolev spaces on complex manifolds

TL;DR

The work introduces higher complex Sobolev spaces on local domains and compact Kähler manifolds, equipping them with quasinorms and related capacities to study complex-analytic regularity. It proves Moser–Trudinger inequalities with exponents growing like , establishes psh majorants to bound , and derives both local and global exponential integrability results. The paper then connects these spaces to the complex Monge-Ampère operator, showing inclusions into energy classes and (global) and to the Cegrell–Blocki domain (local), via inductive energy estimates. These results position as a robust framework that bridges Sobolev-type regularity, pluripotential theory, and Monge-Ampère equations, with implications for variational approaches and geometric analysis on complex manifolds. Overall, the work demonstrates that higher complex Sobolev regularity enhances Moser–Trudinger type control and aligns with established Monge-Ampère energy frameworks, enabling new tools for complex dynamics and Kähler geometry.

Abstract

We study higher complex Sobolev spaces and their corresponding functional capacities. In particular, we prove the Moser-Trudinger inequality for these spaces and discuss some relationships between these spaces and the complex Monge-Ampère equation.
Paper Structure (13 sections, 26 theorems, 148 equations)

This paper contains 13 sections, 26 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Higher complex Sobolev space
  3. Quasinorms and compactness
  4. Examples
  5. Density theorems
  6. Functional capacities
  7. Moser-Trudinger inequalities
  8. Set up
  9. Estimates
  10. Bounding by plurisubharmonic functions
  11. Complex Monge-Ampère equation
  12. Global setting
  13. Local setting

Key Result

Theorem 1.2

Let $q\geq 1$, $\alpha \in[1,2^q)$ and $K$ be a compact subset of the unit ball $\mathbb{B}$ of $\mathbb{C}^k$. Let $v_1,\ldots,v_k$ be psh functions on $\mathbb{B}$ which are Hölder continuous of Hölder exponent $\beta$ for some $\beta \in (0,1)$ with $\|v_j\|_{\mathscr{C}^\beta} \leq 1$ for $1\leq In particular, there exist strictly positive constants $c_1$ and $c_2$ depending on $K$ and $\alpha

Theorems & Definitions (57)

  • Definition 1.1: Higher complex Sobolev spaces
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 47 more