High-contrast homogenization: Provider.Transport.WhitneySquareWeights #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The square weights of a filling row #
The centered part of the principal cell sum of the grid transport is estimated
row by row, and each row contributes through the square weight
(ฮฃ_{V โ ๐ฑ_r(W;q)} (|V|/|W|)ยฒ)^{1/2}: the finite-range matrix averaging lemma
turns a first-moment row bound into a bound on a centered matrix sum only at
the cost of replacing the row's weights by their โยฒ norm. This file proves
the two square weights the transport uses, the square-weight bounds for the
boundary and bulk cells.
Both are pure counting, and both follow from a single observation: the cells of
one row of the filling are aligned cells of the same scale, so they all have
the same relative volume, and the sum of squares is that common weight times the
row's first-moment sum. The boundary row therefore pays the cross-grid row
bound e.two.grid.whitney.volumes, of size 3^{r-j}, once, and the
common weight, of size 3^{d(r-j)}, once, for the printed exponent
(d+1)(r-j). The bulk row pays only the common weight, the first-moment sum of
a disjoint family inside the target being at most one, for the printed exponent
-dฮป_j.
The one quantitative ingredient beyond the rows is the determinant ratio of the
two grids: the common weight of a scale-r row is |det q|/|det p| ยท 3^{d(r-j)},
and the printed constant C(d,K_hop) must absorb the ratio. It does, because a
linear image of the unit cube fits inside a cube of side โd |M|, so
|det M| โค (โd|M|)^d, and the grid ratio dominates |p^{-1}q|. The exhibited
constant is
C(d,K_hop) = (6 d^{3/2} K_hop (โd K_hop)^d)^{1/2},
one constant for both displays.
The determinant of a linear map against its operator norm #
The determinant against the operator norm. The image of the unit cube
under M lies in the coordinate cube of side โd |M|, and the image has volume
|det M|, so |det M| โค (โd|M|)^d. The dimensional factor is the passage from
the Euclidean norm, which the operator norm measures, to the coordinate cube.
The common relative volume of a row #
Every cell of a filling row has the same relative volume, in closed
form: an aligned scale-a cell of the grid q inside the scale-j target cell
of the grid p has relative volume |det q| 3^{ad} / (|det p| 3^{jd}).
The sum of squares of a row is the common weight times the row sum.
This is the whole content of the two square weights: a row of the filling
consists of cells of one scale, so its โยฒ norm is controlled by its โยน norm
with the common weight as the loss.
The common weight of a row against the grid ratio. The relative volume
of a scale-a cell inside the scale-j target is the determinant ratio of the
two grids times 3^{d(a-j)}, and the determinant ratio is at most
(โd K_hop)^d.
The two square weights #
The boundary square weight in squared form. For a row below the
starting scale the first-moment sum carries the cross-grid decay 3^{a-j} and
the common weight carries 3^{d(a-j)}, for the printed exponent (d+1)(a-j).
The bulk square weight in squared form. A row is a disjoint family inside the target, so its first-moment sum is at most one and only the common weight survives.
The printed displays #
The square-weight bound for the boundary cells. At a boundary source
scale the โยฒ norm of the relative volumes of a filling row is at most
C(d,K_hop) 3^{(d+1)(r-j)/2}, with
C(d,K_hop) = (6 d^{3/2} K_hop (โd K_hop)^d)^{1/2}.
A boundary scale is one strictly below the scale at which the filling of the
target starts; for the filling of a scale-j target with buffer ฮป_j that is
exactly the printed condition r < j - ฮป_j.
The square-weight bound for the bulk cells. At the bulk source scale
j - ฮป the โยฒ norm of the relative volumes of a filling row is at most
C(d,K_hop) 3^{-dฮป/2}, with the same constant
C(d,K_hop) = (6 d^{3/2} K_hop (โd K_hop)^d)^{1/2}
as the boundary display. Unlike the boundary rows this one is unconditional in the scale: it needs only disjointness inside the target, which is why the row at which the filling starts, carrying no cross-grid decay, is still summable.