On decoupling of functions of normal vectors

On decoupling of functions of normal vectors. II

“…The proof of Theorem 1.3 follows the framework used in [3]. We need to introduce notations used in [3] and borrowed from [4].…”

“…It is well-known that P n 1 (Ȳ α ) and P n 2 (Ȳ β ) are orthogonal whenever n 1 = n 2 (see e.g. [3]). Consequently, to prove (1) for arbitrary ϕ α it suffices to prove it for φ α (Ȳ α ) = P n (Ȳ α ) for each n = 0, 1, .…”

“…Using Lemma 3.1 from [3] (for the case of diagonal R, see also formula (3.5) in [3] By standard linear algebra argument we have that the right hand side is bounded by σ 0 N α=1 P n ( Ȳα ) 2 2 , where σ 0 is the largest eigenvalue of the matrix S * . We conclude that for a fixed n the estimate (1) holds for φ α ( Ȳα ) = P n ( Ȳα ), α = 1, .…”