Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantitative Hölder Estimates for Even Singular Integral Operators on Patches

Francisco Gancedo, Eduardo Garcia-Juarez

TL;DR

The paper develops a constructive framework to obtain $\dot{C}^{\sigma}$ bounds for even Calderón-Zygmund kernels acting on the characteristic function of a $C^{1+\sigma}$ domain, clarifying how boundary regularity and the arc-chord condition control boundary singularities. By recasting the problem through a boundary operator with an odd kernel on $\partial D$ and performing a meticulous, multi-regime analysis (on/beyond the boundary, near the boundary in nearly normal and tangential directions, and far from the boundary), the authors derive a linear-in-norm bound that explicitly depends on $|\partial D|$, $\|D\|_*$, $\|D\|_{Lip}$, and $\|D\|_{\dot{C}^{1+\sigma}}$. The main result shows that $T(1_D)$ belongs to piecewise $\dot{C}^{\sigma}$ across $\overline{D}$ and $\mathbb{R}^n\setminus D$, with quantitative estimates, enabling applications to free-boundary incompressible Navier-Stokes problems with viscosity contrasts. This tight control advances the understanding of how interface regularity propagates through singular integral operators and supports global-in-time regularity analyses in fluid dynamics contexts.

Abstract

In this paper we show a constructive method to obtain $\dot{C}^σ$ estimates of even singular integral operators on characteristic functions of domains with $C^{1+σ}$ regularity, $0<σ<1$. This kind of functions were shown in first place to be bounded (classically only in the $BMO$ space) to obtain global regularity for the vortex patch problem [5, 2]. This property has then been applied to solve different type of problems in harmonic analysis and PDEs. Going beyond in regularity, the functions are discontinuous on the boundary of the domains, but $\dot{C}^σ$ in each side. This $\dot{C}^σ$ regularity has been bounded by the $C^{1+σ}$ norm of the domain [8, 14, 16]. Here we provide a quantitative bound linear in terms of the $C^{1+σ}$ regularity of the domain. This estimate shows explicitly the dependence of the lower order norm and the non-self-intersecting property of the boundary of the domain. As an application, this quantitative estimate is used in a crucial manner to the free boundary incompressible Navier-Stokes equations providing new global-in-time regularity results in the case of viscosity contrast [12].

Quantitative Hölder Estimates for Even Singular Integral Operators on Patches

TL;DR

The paper develops a constructive framework to obtain bounds for even Calderón-Zygmund kernels acting on the characteristic function of a domain, clarifying how boundary regularity and the arc-chord condition control boundary singularities. By recasting the problem through a boundary operator with an odd kernel on and performing a meticulous, multi-regime analysis (on/beyond the boundary, near the boundary in nearly normal and tangential directions, and far from the boundary), the authors derive a linear-in-norm bound that explicitly depends on , , , and . The main result shows that belongs to piecewise across and , with quantitative estimates, enabling applications to free-boundary incompressible Navier-Stokes problems with viscosity contrasts. This tight control advances the understanding of how interface regularity propagates through singular integral operators and supports global-in-time regularity analyses in fluid dynamics contexts.

Abstract

In this paper we show a constructive method to obtain estimates of even singular integral operators on characteristic functions of domains with regularity, . This kind of functions were shown in first place to be bounded (classically only in the space) to obtain global regularity for the vortex patch problem [5, 2]. This property has then been applied to solve different type of problems in harmonic analysis and PDEs. Going beyond in regularity, the functions are discontinuous on the boundary of the domains, but in each side. This regularity has been bounded by the norm of the domain [8, 14, 16]. Here we provide a quantitative bound linear in terms of the regularity of the domain. This estimate shows explicitly the dependence of the lower order norm and the non-self-intersecting property of the boundary of the domain. As an application, this quantitative estimate is used in a crucial manner to the free boundary incompressible Navier-Stokes equations providing new global-in-time regularity results in the case of viscosity contrast [12].

Paper Structure

This paper contains 8 sections, 2 theorems, 276 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Proof of Theorem \ref{['MainTheorem']}
  4. Regularity on the boundary.
  5. Regularity near the boundary.
  6. Regularity in nearly normal direction:
  7. Regularity in nearly tangential direction:
  8. Regularity far from the boundary

Key Result

Theorem 2.2

Assume $D$ is a bounded domain of class $C^{1+\sigma}$, $0<\sigma<1$. Then, the operator Soperator maps boundedly $\dot{C}^{\sigma}(\partial D)$ into $\dot{C}^{\sigma}(\overline{ D})\cup \dot{C}^{\sigma}(\mathbb R^n\smallsetminus D)$. Moreover, the following bound holds with $P$ a polynomial function depending on $S$, and $C=C(n,\sigma)$.

Theorems & Definitions (3)

  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3