PrincipalResponseMomentsOperator #
Ported from scottnarmstrong/Superdiffusion at 7f20476385d4af418f110303085ad0740c9eba46. Module imports, support namespace and local option directives adapted for Lean Pool.
Provider: sub-step (iv) of the principal response, the operator-norm display #
Source display in ABK26, inside Step 3 of the proof of
l.approximate.recurrence.formula (label):
E[ | bfAhom_{m-1}^{-1/2} bfA_m(cu_n) bfAhom_{m-1}^{-1/2} |^8 ]^{1/8}
<= C E[ ( shom_{m-1}^{-1} Lambda_{gamma,1}(cu_m)
+ shom_{m-1} lambda_{gamma,1}(cu_m) )^8 ]^{1/8}
<= C gamma^{-1} ,
introduced by "by Proposition p.cg.ellipticity.bounds (label) and
e.ellipticities.monotone.ordered (label)".
What is proved here, and what is not #
This module supplies the first inequality of that display, and supplies it
pathwise -- as a deterministic inequality between two functions of the
coefficient field -- which is strictly stronger than the printed inequality
between eighth moments. The second inequality, the numerical bound
C gamma^{-1} for the eighth moment of the right-hand side, is not proved
here: it is the content of p.cg.ellipticity.bounds at s = gamma, q = 1,
and nothing below invokes that proposition or any consequence of it.
The second summand #
Every statement below carries the corrected lambda^{-1}; the printed lambda
occurs nowhere.
The reading of | . | #
For the symmetric positive block matrix
M = bfAhom^{-1/2} bfA(U; a) bfAhom^{-1/2} the manuscript's |M| is the
Euclidean operator norm, which for such a matrix is exactly the supremum of the
doubled quadratic form X . M X over the unit sphere. The statements below
are that supremum bound, written as the homogeneous inequality
X . M X <= c (X . X) for every doubled load X; conjugatedCoarseBlockMatrix
is the matrix M itself, so no reading of the display is left implicit.
Route #
The mechanism is one line of block algebra rather than an estimate. Writing P = bfAhom^{-1/2} X and Q = bfAhom^{1/2} X, the pairing P. Q collapses to
X. X because the two legs of bfAhom^{\pm 1/2} are reciprocal scalars on
reciprocal blocks, so the splitting identity e.bfcoarsegrainedmatrices
(label; the double-variable bfJ it is about is e.bfJfull.var, label)
bfJ(U, P, Q) = 1/2 P . bfA(U) P + 1/2 Q . bfA_*^{-1}(U) Q - P . Q
together with the positivity of bfA_*^{-1}(U) gives
X . bfAhom^{-1/2} bfA(U) bfAhom^{-1/2} X <= 2 bfJ(U, P, Q) + 2 (X . X) ,
and the already-proved sharp one-cube estimate of e.J.by.f (label) for a
scalar comparator bounds bfJ at a unit probe by shom^{-1}|b(U)| + shom|sigma_*^{-1}(U)| - 2. The two one-cube norms are then replaced by
Lambda_{s,q} and lambda^{-1}_{s,q} by e.ellipticities.monotone.ordered,
and the localization cube cu_n is traded for cu_m by
e.bound.one.cube.by.lambdas (label).
Main results #
conjugatedCoarseBlockMatrix-- the matrixMof the display.blockVecDot_conjugatedCoarseBlockMatrix_le_weightedCoarseEllipticity-- the display's|M|bound by the two one-cube norms.blockVecDot_coarseBlockMatrix_le_weightedCoarseEllipticity-- the same statement read as a domination ofP . bfA(U) P.coarseEllipticity_le_multiscaleEllipticity,blockVecDot_coarseBlockMatrix_le_weightedMultiscaleEllipticity-- the same display once the two one-cube norms are replaced byLambda_{s,q}andlambda^{-1}_{s,q}on the cube itself.
The conjugated matrix of the display #
The matrix bfAhom^{-1/2} bfA(U; a) bfAhom^{-1/2}.
bfAhom is the deterministic scalar block diagonal annealedLimitBlock, so
its inverse square root is annealedLimitBlockInvSqrt; the conjugation is the
one printed inside the norm.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The conjugated matrix acts by applying the inverse square root, then the coarse block matrix, then the inverse square root again.
The pairing collapse #
The two legs of bfAhom^{\pm 1/2} are reciprocal on reciprocal blocks, so
the pairing of bfAhom^{-1/2} X with bfAhom^{1/2} X is the plain squared
length of X. This is why the - P . Q term of the splitting identity
contributes the exact constant - 2 at a unit probe.
The unit probe #
The display at a unit probe: for X . X = 1,
X . bfAhom^{-1/2} bfA(U; a) bfAhom^{-1/2} X
<= 2 shom^{-1}|b(U;a)| + 2 shom|sigma_*^{-1}(U;a)| - 2 .
The - 2 is retained because it is exact, not because it is needed.
The homogeneous form #
The display, in operator-norm form. For every doubled load X,
X . bfAhom^{-1/2} bfA(U; a) bfAhom^{-1/2} X
<= ( 2 shom^{-1}|b(U;a)| + 2 shom|sigma_*^{-1}(U;a)| - 2 ) (X . X) ,
which for the symmetric positive matrix on the left is exactly the manuscript's
| bfAhom^{-1/2} bfA(U;a) bfAhom^{-1/2} | <= 2 shom^{-1}|b| + 2 shom|sigma_*^{-1}| - 2.
The load reading: bfAhom^{1/2} undoes the conjugation #
Applying bfAhom^{-1/2} after bfAhom^{1/2} is the identity on doubled
loads.
The squared length | bfAhom^{1/2} P |^2. It is the shom-weighted pair
shom |p|^2 + shom^{-1} |q|^2, i.e. exactly the integrand of the sub-step (ii)
display.
The display, read on a load. The conjugation is undone by writing the load
as bfAhom^{-1/2} applied to a doubled vector: for every load Y,
Y . bfA(U; a) Y
<= ( 2 shom^{-1}|b(U;a)| + 2 shom|sigma_*^{-1}(U;a)| - 2 )
( shom |Y_1|^2 + shom^{-1} |Y_2|^2 ) ,
whose right-hand factor is | bfAhom^{1/2} Y |^2. This is the form in which
the display is consumed: it is the pointwise domination X <= B . V^2 of the
Hoelder split, with B the operator norm and V the normalized load length.
The corrected display on one triadic cube #
e.ellipticities.monotone.ordered (label) at the two ends of its chain, in the
form the display needs: the two one-cube norms are dominated by Lambda_{s,q}
and by lambda^{-1}_{s,q}.
Y . bfA(R; a) Y
<= 2 ( shom^{-1} Lambda_{s,q}(R; a) + shom lambda^{-1}_{s,q}(R; a) )
( shom |Y_1|^2 + shom^{-1} |Y_2|^2 ) .