Quantitative Hölder Estimates for Even Singular Integral Operators on Patches

Abstract

In this paper we show a constructive method to obtain new C σ estimates of even singular integral operators on characteristic functions of domains with C 1`σ regularity, 0 ă σ ă 1. This kind of functions were shown in first place to be bounded (classically only in the BM O space) to obtain global regularity for the vortex patch problem [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 C σ in each side. This C σ regularity has been bounded polynomially by the C 1`σ norm of the domain [13].Here we provide a sharper bound of type L log L in terms of the C 1`σ regularity of the domain. This new estimate shows explicitly the dependence of the lower order norm and the non-selfintersecting property of the boundary of the domain. As an application, this new 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 [10].