Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fine properties of functions in complex Sobolev spaces

Ngoc Cuong Nguyen

Abstract

We study comprehensively local properties of functions in complex Sobolev spaces on a bounded open subset of $\mathbb{C}^n$. The main tool is the corresponding functional capacity for the space which is inspired by the global one due to Vigny (2007). An inequality between this capacity and the Bedford-Taylor capacity for plurisubharmonic functions is proved, which is sharp as far as the exponents are concerned. Moreover, it is shown that the functional capacity is a Choquet capacity. The Alexander-Taylor type inequality for the capacity is also proved. This allows us to strengthen the results in the works of Dinh, Marinescu and Vu (2023), Vigny and Vu (2024). Lastly, the Moser-Trudinger type inequality in this space is characterized by the volume-capacity inequality.

Fine properties of functions in complex Sobolev spaces

Abstract

We study comprehensively local properties of functions in complex Sobolev spaces on a bounded open subset of . The main tool is the corresponding functional capacity for the space which is inspired by the global one due to Vigny (2007). An inequality between this capacity and the Bedford-Taylor capacity for plurisubharmonic functions is proved, which is sharp as far as the exponents are concerned. Moreover, it is shown that the functional capacity is a Choquet capacity. The Alexander-Taylor type inequality for the capacity is also proved. This allows us to strengthen the results in the works of Dinh, Marinescu and Vu (2023), Vigny and Vu (2024). Lastly, the Moser-Trudinger type inequality in this space is characterized by the volume-capacity inequality.
Paper Structure (19 sections, 66 theorems, 364 equations)

This paper contains 19 sections, 66 theorems, 364 equations.

Table of Contents

  1. Introduction
  2. Definitions and elementary properties
  3. Complex Sobolev spaces
  4. Basic examples of functions
  5. Approximation
  6. Compactness and completeness
  7. Bedford-Taylor capacity for plurisubharmonic functions
  8. Basic integral inequalities
  9. A functional capacity
  10. Quasi-continuity
  11. Capacitability
  12. Embedding into Dirichlet spaces
  13. Dirichlet spaces associated with a closed positive current
  14. Capacity in the Dirichlet spaces
  15. Exponential integrability
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\Omega \subset \subset \mathbb{C}^n$ be a strictly pseudoconvex domain. Moreover, the above inequalities are sharp as far as the exponents are concerned.

Theorems & Definitions (145)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Corollary 2.4
  • Example 2.5
  • ...and 135 more