Results on metric Markov cotype, cuts in the sequence space, and Lipschitz extension

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.

• Hilbert space has metric Markov cotype two — Proposition 3.4. For a stochastic matrix A reversible for pi, t >= 1 and points x_i of a real Hilbert space, the Cesaro averages y_i = sum_j (M_t)_ij x_j satisfy sum pi_i |x_i - y_i|^2 + t E_A(y) <= 5 E_(M_t)(x). So N_2 <= sqrt 5.
• Energies along a stationary chain are subadditive in the square root — Lemma 4.1. For a stochastic matrix A with stationary vector pi and points of a metric space, D(s) = sum pi_i (A^s)_ij d(x_i,x_j)^2 satisfies sqrt D(r+s) <= sqrt D(r) + sqrt D(s).
• Geometric against Cesaro averages — Proposition 4.3. For a stochastic matrix A with stationary vector pi, points of a metric space and t >= 1: with G_t = (1/(t+1)) sum_(s>=0) (t/(t+1))^s A^s and M_t = (1/t) sum_(s=1..t) A^s, E_(G_t)(x) <= 28 E_(M_t)(x). Conversely E_(M_t)(x) <= 2e E_(G_t)(x) (Exercise 5.4).
• Cut representation of finitely many points of l_1 with a reconstruction map — Proposition 1.2. For x_1..x_n in l_1, not all equal, there are a finite set of cuts B with weights w_B > 0, vertices z_i in {0,1}^B and an affine map T from R^B to l_1 with T(z_i) = x_i, |x_i - x_j|_1 = sum w_B |z_i(B) - z_j(B)| = sum w_B (z_i(B) - z_j(B))^2, and |T(u) - T(u')|_1 <= sum w_B |u_B - u'_B|.
• The flat inequality for the cubic — Proposition 3.3. On the cube [0,1]^B with weights w, let Phi apply 3r^2 - 2r^3 in every coordinate. For u, u' in the cube and a vertex e, with a = u - e and delta = u' - u: |Phi(u') - Phi(u)|_(1,w)^2 <= 108 (|u' - e|_H^4 - |u - e|_H^4 - 4 |a|_H^2 <a, delta>_H).
• Mean-zero displacements — Corollary 4.1. If u = sum rho_l u^l with points of the cube and weights rho_l >= 0 of sum 1, then sum rho_l |Phi(u^l) - Phi(u)|_(1,w)^2 <= 108 (sum rho_l F_e(u^l) - F_e(u)) for every vertex e, F_e(u) = |u - e|_H^4.
• The cotype inequality for l_1 — Theorem 3.1. For every stochastic matrix A with a stationary vector pi (reversibility is not needed), t >= 1 and x_1..x_n in l_1 there are y_1..y_n in l_1 with sum pi_i |x_i - y_i|_1^2 + t sum pi_i a_ij |y_i - y_j|_1^2 <= 3024 sum pi_i (M_t)_ij |x_i - x_j|_1^2. So N_2(l_1) <= 12 sqrt 21 (Corollary 3.2). No hypothesis.
• The new points are means of medians of three — Proposition 4.2. y_i = sum over j_1, j_2, j_3 of G_(i j_1) G_(i j_2) G_(i j_3) med(x_(j_1), x_(j_2), x_(j_3)), the coordinatewise median, where G is the geometric average of the powers of A with ratio t/(t+1).
• Complex l_1 — Corollary 1.2. The complex space l_1 has metric Markov cotype two with the constant 12 sqrt 42.
• The cotype inequality for L_1(mu) — Theorem 2.1. For every measure space, real L_1(mu) satisfies the inequality of Theorem 3.1 of the third lesson with the same constant 3024, for every stationary matrix. No hypothesis.
• Hilbert space has Markov type two with the constant 1 — Proposition 3.2. For A reversible for pi and points of a real Hilbert space: sum pi_i (A^t)_ij |x_i - x_j|^2 <= t sum pi_i a_ij |x_i - x_j|^2.
• l_1 is the dual of c_0 — Lemma 4.1. a -> (x -> sum a(k) x(k)) is a linear isometry of l_1 onto the dual of c_0.
• Extension of Lipschitz maps from Hilbert space into l_1 — Corollary 5.1. Assume (MN) (the extension theorem of Mendel and Naor; not proved in the programme). There is K such that every Lipschitz map from a subset of a real Hilbert space into l_1 extends to the whole space with Lipschitz constant at most K times the original one.

Reading order

Four lessons. The first gives the definition and two general facts; the second prepares the two tools; the third proves the theorem, which needs no hypothesis; the fourth gives consequences. Only the extension of Lipschitz maps in the fourth lesson rests on a named hypothesis, the extension theorem of Mendel and Naor, which is not proved in the programme.