The whole-carrier pressure-gradient time mass of prop:bootstrap #
The B slot of prop:bootstrap asks for the actual clipped L^{6/5} time mass
of a selected weak pressure gradient on the origin carrier B_{R₁}, measured
against a coefficient carrying no velocity or gradient Morrey datum. This file
proves that estimate at the two exponent/radius triples of the endgame, above
one absolute Calderón–Zygmund threshold.
The route is: transfer the binder's field to the interior measurable weak
gradient of eq:pressure-gradient-morrey, cover the carrier by the fixed finite
lattice of collars of radius 1/8, and add the collar estimates.
The whole-carrier pressure-gradient time mass of prop:bootstrap #
The B slot of prop:bootstrap asks for the actual clipped L^{6/5} time mass
of the selected weak pressure gradient on the origin carrier B_{R₁}, measured
against the numerical coefficient
(|C_CZ| + 1) * (|R₀| + |R₁| + |ε| + 1) * max 1 ((2R₁/(1-R₁)) ^ θ),
where θ = 5 (1 - (6/5) / min ((1/τ + 8/25)⁻¹) q). That coefficient carries
neither velocity nor gradient Morrey data, so the estimate has to come from the
ε data alone.
This file records the numerical fact that makes the coefficient usable: any
absolute mass below a threshold Cstar is paid by the coefficient once
Cstar ≤ C_CZ, on the whole admitted parameter range 5/2 < q,
25/3 ≤ τ ≤ 25, 0 < R₁ < 3/4.
Above an absolute Calderón–Zygmund threshold the B-slot coefficient of
prop:bootstrap pays any ε-linear mass whose absolute factor is below the
threshold.
Any weak slice derivative of the pressure on an origin carrier agrees
almost everywhere with the interior measurable weak gradient of
eq:pressure-gradient-morrey on the ball of radius 7/8.
The fixed lattice of collars covers every origin carrier of radius at most
11/16, so the carrier mass is at most the sum of the cell masses.
The absolute harmonic remainder constant of eq:pressure-gradient-decomposition,
supplied by the interior estimate for the harmonic pressure part.
Equations
Instances For
The remainder constant is nonnegative and majorizes the smooth remainder gradient on every collar half ball.
The whole-carrier coefficient of prop:bootstrap: the fixed collar count
times the collar coefficient at radius 1/8.
Equations
Instances For
The absolute Calderón–Zygmund threshold above which the B slot of
prop:bootstrap is paid by its own coefficient.
Instances For
The whole-carrier pressure-gradient time mass. Any weak slice
derivative of the pressure on an origin carrier of radius at most 11/16 has
6/5 time mass at most an explicit absolute multiple of ε + 1.
The collar coefficient is finite at every positive radius.
The whole-carrier coefficient is finite.
The threshold is the real value of the finite whole-carrier coefficient.
The B slot of prop:bootstrap at the two endgame instances. Above the
absolute threshold bslotThresholdB, the actual clipped L^{6/5} time mass of
any weak pressure gradient on the origin carrier is paid by the numerical
coefficient of prop:bootstrap, which carries no Morrey datum.