Results on the dyadic triangular form and matrix energies: the trace identity, the children, the Hessian and order comparison, the trace estimate, the energy increase and the theorem, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• The dyadic triangular Hilbert form on L^3 (OpenAI) — Theorem A. For every finite set S of scales and real F_0, F_1, F_2 in L^3(R_+^2), the sum over k in S and over admissible triples I of |L_I(F_0, F_1, F_2)| is at most 40 prod ||F_v||_3. Proved in the last lesson.
• Local forms as traces — Lemma 1.1. For step functions on atoms of level m and an admissible triple of scale k >= m + 1, L_I = 2^(2k) tr(D_0 W_0 D_1 W_1 D_2 W_2).
• The four children of an admissible triple — Lemma 1.2. The admissible children have sign triples with product 1; conditionally on one sign, each other sign is uniform; every admissible triple has exactly one admissible parent.
• One scale — Lemma 1.3. For L^3 inputs, the sum over the admissible triples of one scale of |L_I| is at most prod ||F_v||_3.
• Hessian of tr(P^(3/2)) — Lemma 2.1. d Phi_P(K) = (3/2) tr(sqrt P K) and d^2 Phi_P(K, K) = b_P(K) = (3/2) sum K_ij^2/(sqrt p_i + sqrt p_j); Phi is convex on positive semidefinite matrices.
• Monotonicity of the inverse and the square root — Lemma 2.2. For positive definite X <= Y, Y^{-1} <= X^{-1} and sqrt X <= sqrt Y.
• Order comparison of Hessians (OpenAI) — Lemma 2.3. If P <= aQ with a >= 1, every K supported on P is supported on Q and b_P(K) >= a^(-1/2) b_Q(K).
• Midpoint gap — Corollary 2.4. If -T <= U <= T, then (Phi(T + U) + Phi(T - U))/2 - Phi(T) >= b_T(U)/(2 sqrt 2).
• Mixed trace estimate (OpenAI) — Proposition 2.5. For maps W_v around a triangle and symmetric contractions D_v, |tr(D_0 W_0 D_1 W_1 D_2 W_2)| <= (5/3) sum_v b_{T_v}(U_v), with row stars R_v, T_v = R_v^T R_v and U_v = R_v^T D_v R_v; the constant does not depend on the dimensions.
• Local energy increase (OpenAI) — Proposition 3.1. The average energy of the four children exceeds the energy of an admissible triple by delta(I) >= 0, and |tau(I)| <= (10 sqrt 2/3) delta(I).
• Total energy and telescoping — Lemma 3.2. J_k = sum over admissible I of 2^(2k) e(I) is at most 2 sqrt 2 sum ||F_v||_3^3, and the weighted energy increases at scale k add up to J_(k-1) - J_k.

Reading order

Read the lessons in order. The second lesson, on matrices, can be read on its own; the third combines it with the dyadic structure of the first.