  In this paper we show a constructive method to obtain $\dot{C}^\sigma$
estimates of even singular integral operators on characteristic functions of
domains with $C^{1+\sigma}$ regularity, $0<\sigma<1$. 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 $\dot{C}^{\sigma}$ in each side. This
$\dot{C}^{\sigma}$ regularity has been bounded by the $C^{1+\sigma}$ norm of
the domain [8, 14, 16]. Here we provide a quantitative bound linear in terms of
the $C^{1+\sigma}$ 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].
