The every-cell Morrey bound in its two regimes #
The every-cell Morrey bound of a field carried by a measurable set has two
regimes. Below the fixed scale r₀ the cell of the field is controlled by the
slice growth bound, applied at the cell's own radius. At or above r₀ the
cell power integral is bounded by the whole-carrier integral, which is a
constant bound of the growth form at the cost of the explicit scale factor
r₀ ^ (-(5 (1 - P/κ))). Combining the two gives a single every-cell estimate
whose constant is the sum of the two regime constants, and hence a bound on the
Morrey seminorm itself.
A single uniform argument across all radii does not exist: the slice bound is available only while the cell is small relative to its distance to the carrier boundary, so the two regimes must be kept separate and then glued.
The every-cell growth bound in its two regimes #
A cell of the pressure-gradient carrier is controlled by the slice estimate at
its own scale only when its ball still fits inside the carrier at that scale;
a cell that is large relative to its distance to the carrier boundary is not
reached by the slice estimate at all. The growth bound A · r ^ (5 (1 - P/κ))
of the parabolic Morrey class therefore has two regimes.
Below a fixed scale r₀ the bound is the slice bound at the cell's own radius.
At or above r₀ the cell power integral is bounded by the whole-carrier
integral, and because the growth exponent 5 (1 - P/κ) is nonnegative for
P ≤ κ, that constant bound is itself of the required form, at the cost of the
explicit scale factor r₀ ^ (-(5 (1 - P/κ))).
The two regimes are combined here into a single every-cell statement whose constant is the sum of the two. Attempting one uniform argument across all radii is what makes a growth clause unsatisfiable.
A field vanishing off a measurable carrier has every cell power integral bounded by its integral over that carrier.
The large-cell regime. At or above the scale r₀ a constant bound on the
cell power integral is already of the Morrey growth form, with the explicit
scale factor.
The every-cell growth bound of one field, in its two regimes: the slice
bound at the cell's own scale below r₀, and the whole-carrier integral with
the explicit scale factor at or above r₀.
A cell whose power integral satisfies the growth bound at its own radius is bounded by the power of the growth constant: the radius weight of the Morrey cell cancels the radius factor carried by the bound.
The every-cell Morrey bound of a field carried by a measurable set, in its
two regimes: the constant is the power of the sum of the slice constant and the
whole-carrier integral rescaled by r₀ ^ (-(5 (1 - P/κ))).
Restricting a field to a carrier can only decrease a cell power integral.
A cell that misses the carrier carries no mass of the restricted field. This is the trivial half of the small-cell regime: only cells meeting the carrier need the slice estimate.