Exponential mixing of all orders on Kähler manifolds: (quasi-)plurisubharmonic observables

The paper proves exponential mixing of all orders for d.s.h. observables with the sharp 1/2 exponent in the gap, precisely as stated in Theorem 1.1, using a product-space construction on X×X, carefully designed auxiliary functions Φ±, super-potential regularity, and a key convergence estimate (Proposition 3.6); it then reduces general d.s.h. observables to bounded quasi-p.s.h. ones via a quantitative decomposition (Lemma 2.4), completing the proof with uniform control of constants . By contrast, the model’s solution misidentifies the diagonal current [Δ] as a Green current for a “suitable product automorphism” (in the paper, the Green (k,k)-currents on X×X are T+⊗T− and T−⊗T+, and normalized iterates of [Δ] converge to T− under F=(f,f^{-1}), not to [Δ] itself) . It also asserts one may take the threshold δ′=δ0; the paper shows δ′ must strictly exceed δ0 by the explicit formula δ′=d_p^{1/(1+λ0)} δ0^{λ0/(1+λ0)} determined by the Hölder exponent λ0 of the super-potential of T+∧T−, which is essential for the final contraction estimate . Finally, the model’s core “Cauchy–Schwarz–type pairing” mechanism for producing the 1/2 exponent is neither developed nor needed in the paper, where the 1/2 exponent arises from a half-iterate F^{n/2} and the Φ± device (Lemma 3.7) . The model also cites this very paper as a dependency, creating a circular justification.