The collar slice bound for the whole-carrier pressure gradient #
On a collar Q(z, ρ) inside the unit data cylinder of thm:A, every weak
slice derivative of the pressure on the half ball B(z₁, ρ/2) is, by
eq:pressure-gradient-decomposition, the sum of three completed Riesz terms built from
the localized source, one smooth remainder gradient, and three completed Riesz
terms built from the localized force. This file turns that identification
into a slice inequality between L^{6/5} norms in which the localized source
and the localized force have been replaced by majorants that no longer depend
on the component index.
No velocity or gradient Morrey datum enters any estimate in this file.
The localized force slice is dominated in L^{6/5} by the force slice on
the collar ball, uniformly in the component index.
The collar slice bound of eq:pressure-gradient-decomposition. On the half
ball of a collar inside the domain, every weak slice derivative of the pressure
has L^{6/5} norm at most three times the Calderón–Zygmund constant applied to
the centred source majorant, plus the smooth remainder majorant weighted by the
half-ball volume, plus three times the same constant applied to the force slice
norm on the collar ball.
The 6/5 time mass of the force slice norms on a collar costs the collar
volume in addition to the unit data of thm:A.
A slice bound of the shape produced by eq:pressure-gradient-decomposition turns
three separate time masses into one, with the numerical factor of the
three-term power split.
The explicit absolute coefficient of the collar time mass of
prop:bootstrap. Every factor is a fixed function of the collar radius and
of the harmonic remainder constant; none depends on the force exponent, on the
Morrey exponent, on the radii of the carrier, on the data size, or on the
solution.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The collar time mass of the weak pressure gradient. On a collar whose
doubled cylinder stays inside the unit data cylinder of thm:A, the 6/5 time
mass of any weak slice derivative of the pressure on the collar half ball is at
most an explicit absolute multiple of ε + 1.