Ugly #
Ported from scottnarmstrong/Superdiffusion at 7f20476385d4af418f110303085ad0740c9eba46. Module imports, support namespace and local option directives adapted for Lean Pool.
The five-term per-cube ("ugly") estimate on the good event #
ABK26, Section 4.1, (e.ugly.estimate.for.J). The manuscript takes the four
sensitivity terms of e.ugly.estimate.for.J.pre and collapses every lambda-
and gradient-factor against the two good-event budgets
sigmaBar_{n-2} lambda_{gamma,2}^{-1}(z+cube_n; a_{n-2}) 1_G <= C 3^{gamma(m-n)}
sigmaBar_{n-2}^{-1} 3^{2n} ||grad(k_L - k_{n-2})||_{W^{1,infty}(z+cube_n)} 1_G
<= C 3^{s(m-n)/4}
(ABK26, in that order)
together with the running-diffusivity facts
sigmaBar_k >= (1/2) cstar^{1/2} gamma^{-1/2} 3^{gamma k} (every k <= m)
sigmaBar_m^{-1} sigmaBar_{n-2} <= C
(the second is ABK26)
and one mean-zero Poincare step. The result is the printed five-term display.
The two hypotheses this module does NOT discharge #
hsens--- the four-term pre-estimate. It is produced, at the Section 2.4 anchor, byAlgsuperdiff.Section4.Provider.Annular.UglyPre.hH--- the Poincare splitHsq <= 2 Lnrm^2 + 2 (Cp * 3^{2m} Gm)^2, i.e. the triangle inequality `k_L - (k_L-k_m){cube_m} - k{n-2} = (k_m - k_{n-2})- (k_L - k_m - (k_L-k_m)_{cube_m})
followed by the mean-zero Poincare inequality oncube_m. The manuscript cites "the Poincare inequality" with no label, so it enters here as a hypothesis, at the amplitudeCp`.
- (k_L - k_m - (k_L-k_m)_{cube_m})
The five-term shape #
The shape of e.ugly.estimate.for.J, with the two displayed weights w = 3^{s(m-n)} and w_gamma = 3^{(s+gamma)(m-n)} left abstract.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The running-diffusivity arithmetic #
kap is the normalization gamma * kap^2 = cstar, i.e.
kap = cstar^{1/2} gamma^{-1/2}, kept squared so that no square root ever
appears.
The inverse form of the relaxed sigmaBar lower bound sigmaBar_k >= (1/4) cstar^{1/2} gamma^{-1/2} 3^{gamma k}, i.e. the shape the Section 3 state
delivers at the shifted index k = n-2 once the exponent is moved to 3^{gamma n} (see the module docstring).
The four per-term patches #
Patch 1. The first sensitivity term, whose growth factor GF1 is
already collapsed against the weight W, becomes the displayed
C 3^{s(m-n)} E^2 term.
Patch 3. The third sensitivity term Cs sigmaBar_m^{-2} X^2 becomes the
displayed gradient-at-n term.
Patch 4. The fourth sensitivity term Cs min{1, X lambda^{-1}}^2 also
becomes the displayed gradient-at-n term, by the lambda-budget
sigmaBar_{n-2} lambda^{-1} <= Cl 3^{gamma(m-n)}.
Patch 2. The second sensitivity term splits three ways: the
sigmaBar-continuity term, the L^2 term, and --- via the Poincare hypothesis
hH --- the displayed gradient-at-m term.
Q * Pn = Pm is the triadic identity 3^{gamma(m-n)} 3^{gamma n} = 3^{gamma m}
and W * Q = Wg is 3^{s(m-n)} 3^{gamma(m-n)} = 3^{(s+gamma)(m-n)}.
The Poincare split #
The hH slot of the assembly, produced from the triangle inequality and one
mean-zero Poincare bound.
H is ||k_L - (k_L - k_m)_{cube_m} - k_{n-2}||_{L-underline^2(z+cube_n)}, A
is ||k_m - k_{n-2}||_{L-underline^2(z+cube_n)}, and B is the mean-zero
fluctuation ||k_L - k_m - (k_L - k_m)_{cube_m}||_{L-underline^2(z+cube_n)},
whose Poincare bound is hpoin (there Y = 3^{2m} Gm).
The Poincare input itself is proved upstream, not owed here: CoarseGraining's
Homogenization.cubeBesovOscillation_two_le_cubeScaleFactor_mul_normalizedW1pSeminorm
is exactly ||u - (u)_Q||_{L-underline^2(Q)} <= C(d) 3^{Q.scale} ||grad u||_{L-underline^2(Q)} on a triadic cube, in the volume-normalized convention
this slice uses.
The four-term to five-term assembly #
e.ugly.estimate.for.J, abstract-weight form.
The four sensitivity terms hsens of e.ugly.estimate.for.J.pre, collapsed
against the two good-event budgets (hlam, and the growth-factor bounds
hG/hG which are where the 3^{s(m-n)/4} gradient budget has already been
absorbed), the two sigmaBar lower bounds and the sigmaBar ratio, plus the
Poincare split hH, give the printed five-term display at the explicit
constant C.
The triadic weights: the growth-factor collapses #
With the lambda- and gradient-budgets already inserted, the first sensitivity
growth factor is dominated by (1 + A)^2 3^{s(m-n)}. At the gapped gauge the
exponent is 2s/(1 - 4 gamma) and the arithmetic needs s/2 + 6 gamma <= 1,
which is what s <= 1/2 and gamma <= 1/8 deliver (with equality at the
corner).
Growth-factor collapse, second exponent. At the gapped gauge the exponent
is 4 gamma/(1 - 4 gamma) and the arithmetic needs gamma + 5 s gamma <= s,
which is what 8 gamma <= s <= 1/2 delivers with room to spare (the left side
is at most (7/16) s).
e.ugly.estimate.for.J, at the printed weights w = 3^{s(m-n)} and
w_gamma = 3^{(s+gamma)(m-n)}.
hlam is the lambda-budget, hshell the G_1 gradient budget, hratio the
sigmaBar_m^{-1} sigmaBar_{n-2} <= C, hH the Poincare split, and hsens the
four-term pre-estimate.
The s-window here is the manuscript's [8 gamma, 1/2]. A consumer that
feeds hsens from the Section 2.4 anchor (see UglyPre) must further restrict
to s <= 1/4; that residual is noted there.