The selected weak pressure gradient on one slice #
Display (3.5) of the pressure-gradient section estimates the weak spatial
gradient of a pressure slice after the local decomposition
p = p₁ + p_har + (p₇ + p₈). The first summand is a coordinate sum of
Newtonian derivative potentials of a divergence-form source V and is
differentiated by the Calderón–Zygmund selection; the second is smooth on the
inner ball and is differentiated classically; the third is differentiated by
its own potential identities.
This file performs the assembly: it produces one Vec3-valued slice field
whose coordinates are locally integrable, which lies in L^{6/5} on the inner
set, which is the coordinate weak gradient of the pressure slice there, and
whose coordinate norms obey the three-term bound of display (3.5).
Each coordinate of a globally L^{6/5} vector field is locally integrable
on every set. This is the local integrability that display (3.5) requires of
the Calderón–Zygmund part of the selected gradient.
The one-slice assembly behind display (3.5) of the pressure-gradient section.
On an open set B the pressure slice p agrees almost everywhere with the
sum of the coordinate Newtonian derivative potentials of a compactly supported
L^{6/5} source V, of a function h which is C¹ on B, and of a
function w carrying its own coordinate weak derivatives gw. The
Calderón–Zygmund selection hP1 supplies the weak gradient of the potential
part together with its L^{6/5} bound. The conclusion is a single field D
whose coordinates are locally integrable on B, which lies in L^{6/5} on
B', which is the coordinate weak gradient of p on B, and which obeys the
three-term bound of display (3.5).
The coordinate sum of source norms appearing in the assembled bound is controlled by the norm of the source on the ball carrying its support, which is the form of the first term of display (3.5). The factor three is the number of spatial coordinates.
The assembled bound rewritten with the source norm taken on the ball carrying the support of the source, which is the first term of display (3.5). The three spatial coordinates are absorbed into the constant.