Arazy's conjecture concerning Schur multipliers: revisited and resolved

Abstract: Let S^r denote the Schatten–von Neumann class and let S_{Ψ_{f,λ}}
be the Schur–Hadamard multiplier whose symbol is the divided-difference
matrix of f along λ. Let 0 < α, r < ∞, let f ∈ C^1([−1, 1]) satisfy f (0) = 0,
|f ′(t)| ≲ |t|α, and let λ ∈ ℓ_r be real with ∥λ∥_{ℓ∞} ≤ 1. We determine the pairs
0 < p, q ≤ ∞ for which S_{Ψ_{f,λ}} : S^q → S^p is bounded for every such f and λ.
Our principal new positive estimates treat q = ∞, 1 and 0 < p < 1. Precisely,
we show that the boundedness holds exactly when
1/p ≤ α/r + min{1, 1/q}.
In particular, this resolves the untreated cases in [Arazy, PAMS, 1982] and
[Potapov, Sukochev, Tomskova, Adv. Math., 2015]. Our method also delivers
a new proof of the main results in just cited papers.