[Paper Review] On supnorm estimates for $\bar\partial$ on infinite type convex domains in $\mathbb C^2$

John Erik Fornæss, Lina Lee+1 more|ArXiv.org|Nov 12, 2009

Nonlinear Partial Differential Equations3 citations

TL;DR

This paper establishes $ L^rown $-norm estimates for the $ arackepsilon $-equation on certain convex domains of infinite type in $ \mathbb{C}^2 $, specifically rounded bidisks with exponential-type boundary behavior. Using a modified Henkin integral kernel and careful $ L^1 $-type estimates, the authors prove that uniform $ L^rown $-bounds hold when the exponent $ \alpha < 1 $, resolving a long-standing open problem for this class of domains.

ABSTRACT

In this paper, we study the $\bar\partial$-equation on some convex domains of infinite type in $\mathbb C^2$. In detail, we prove that supnorm estimates hold for infinite exponential type domains provided the exponent is less than 1.

Motivation & Objective

• To establish uniform $ L^\infty $-norm estimates for solutions to the $ \bar\partial $-equation on convex domains of infinite type in $ \mathbb{C}^2 $, where classical methods fail due to infinite type singularities.
• To extend Henkin's integral formula technique from the bidisc to domains with flat and strongly convex boundary components, particularly those with exponential-type flattening.
• To determine the critical exponent $ \alpha $ for which the Henkin kernel remains absolutely integrable, enabling $ L^\infty $ estimates.
• To analyze the behavior of the $ \bar\partial $-solution operator on domains with Levi-flat and strongly convex boundary parts, especially near infinite-type points.
• To provide a framework for $ L^\infty $-estimates on rounded polydiscs with infinite-type boundary profiles, generalizing results from finite-type and strictly convex domains.

Proposed method

• Adapts Henkin's integral kernel formula for the $ \bar\partial $-equation on the bidisc to domains with a flat face and a rounded, infinite-type boundary component.
• Splits the boundary integral into three parts: flat face (Henkin-type), compact strongly convex part, and a transition region near the infinite-type point.
• Applies integration by parts and change of variables to transfer integrals from the flat face to the curved part, preserving $ L^1 $-type bounds.
• Uses polar coordinates and $ L^p $-duality estimates with $ p_0 > 2 $, $ q_0 < 2 $, to control logarithmic and singular terms arising from the exponential flattening.
• Employs the inequality $ a^p + b^q \gtrsim ab $ for $ 1/p + 1/q = 1 $ to separate singularities in the kernel and ensure integrability when $ \alpha < 1 $.
• Analyzes the behavior of the kernel near the infinite-type point by estimating $ |\ln(r^2 - 1)|^{p_0}/(r^2 - 1)^{p_0\alpha/2} $ in polar coordinates, showing convergence when $ \alpha < 1 $.

Experimental results

Research questions

• RQ1 Can $ L^\infty $-norm estimates be established for the $ \bar\partial $-equation on convex domains in $ \mathbb{C}^2 $ that are of infinite type?
• RQ2 What is the critical exponent $ \alpha $ for which the Henkin integral kernel remains absolutely integrable on domains with exponential-type flattening?
• RQ3 Does the method used for the bidisc extend to domains with a mix of flat, strongly convex, and infinite-type boundary regions?
• RQ4 Can $ L^\infty $-estimates be obtained for rounded bidisks where the boundary is smoothed via an exponential profile with exponent $ \alpha < 1 $?
• RQ5 Is the exponent $ \alpha = 1 $ optimal for $ L^\infty $-estimates, or can the bound be improved?

Key findings

• For the domain $ \Omega = \{ \chi(|z_1|^2) + |z_2|^2 < 4 \} $ with $ \chi(t) = 1 + \exp(-1/(t-1)^{\alpha/2}) $ on $ (1,1+\epsilon) $, $ L^\infty $-estimates for $ \bar\partial u = f $ hold when $ \alpha < 1 $.
• The integral kernel associated with the $ \bar\partial $-solution operator remains uniformly in $ L^1 $ on the transition region when $ \alpha < 1 $, due to controlled logarithmic and power singularities.
• The estimate $ \int_{1<r<1+\epsilon} \frac{|\ln(r-1)|^{p_0}}{(r-1)^{p_0\alpha/2}} dr < \infty $ holds when $ \alpha < 1 $, ensuring integrability of the kernel component near the infinite-type point.
• The method successfully extends to rounded bidisks defined by $ \rho(z) = \chi(|z_1|) + \chi(|z_2|) - 2 + a < 0 $, with $ \chi $ convex and exponentially flat, yielding $ L^\infty $-bounds for $ \alpha < 1 $.
• The critical exponent $ \alpha = 1 $ remains unresolved; the authors show that $ \alpha < 1 $ is sufficient but do not prove it is necessary.
• The analysis confirms that the $ L^\infty $-bound for the solution $ u $ satisfies $ \|u\|_\infty \leq C \|f\|_\infty $ for some constant $ C = C(\Omega) $, valid on the specified infinite-type domains.