Trilinear Hilbert transform: normalizedAbsorption_of_activeFinal #
Blueprint prop:normalized (lines 2827-2904) from eq:active-final: the base reduction
(activeCellReduction, every level), the depth-independent initial levels
(setups_total_le), the active-final bound for the later levels, the level sums
(update_levels_sum) and the cutoff limit (levels_cutoff_sum) give eq:normalized with an
arbitrarily small coefficient of C_N.
Blueprint thm:uniform-local from eq:active-final: every admissible class has q ∈ (2, 3)
with sup_N C_N(q) < ∞.
Blueprint thm:main from eq:active-final for every admissible class: the trilinear Hilbert
transform principal value is bounded L³ × L³ × L³ → L¹ on Schwartz functions and extends
uniquely.
Blueprint eq:active-min and eq:active-final (prop:base line 1622, prop:normalized lines
2885-2888) on the active cells: for every pattern with at least three basic arguments, the
weighted minima over the cells active at δ_k = e^{-k^{C_b+2}} are at most
C η₀^{c_*} (1 + C_N) |R|.
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- One or more equations did not get rendered due to their size.
Instances For
The inactive error of the base reduction at δ_k = e^{-k^{C_b+2}} is eventually at most
e^{-k} per unit length, uniformly over setups with budgets exp(k^{C_b}).
Blueprint prop:base (lines 1655-1660) with eq:active-final: the active-cell minima of the
three patterns, the two-U term η₀ k^{C_b} |R| and the inactive error e^{-k} |R| give the
active-final bound for the whole setup.
Blueprint prop:normalized, lines 2840-2842 ("at each such root restart on the original base summands ... their normalized sizes and rolling bounds hold there"): the level properties pass to the cells below every dyadic sub-root of the root.
Blueprint eq:before-absorption on one root, after the choice of q, α, P (prop:normalized,
lines 2838-2888): the retained-attempt estimate. For every active setup with the level
properties there are pairwise disjoint exceptional roots of total length at most |R|/4 such that
the active-cell minima of each pattern over the cells below no exceptional root are at most
C η₀^{c_*} (1 + C_N) |R|.
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- One or more equations did not get rendered due to their size.
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Blueprint prop:normalized, lines 2840-2843 and 2885-2888: restarting the retained-attempt
estimate at the exceptional roots, whose root packing has factor 4/3, gives eq:active-final
on all active cells.
The cells of an active family containing a cell are among the active cells through its left
endpoint (the path multiplicity M_p of the localization).
Blueprint prop:base (line 1655) and line 2839 (the active path multiplicity for the attempt):
at the activity threshold δ_k = e^{-k^{C_b+2}}, every point lies in at most
exp(C k^{3 C_b + 2}) active cells of a setup with the level properties.
The eventual growth bounds of the parameters (blueprint lines 2869-2888) #
eventually_nat_qpoly_le_exp with a constant factor A ≥ 0.
Eventually M ≤ log k.
The polynomial standing bounds of the retained-attempt estimate at accuracy k, for the
exponent Y = L₄ + 2 k^b of the height and column budget (blueprint lines 2853-2869: all degrees
are absolute): the colour count, vertex bound, fuel, source height and counts are below k^c
(attempt) and k^{cc} (leaf), and the curvature thresholds below k^{cc}.