Higher order perturbation estimates in quasi-Banach Schatten spaces through wavelets

Let n ∈ ℕ_≥1. Let 1 ≤ p1,…,p_n < ∞ and set the Hölder combination p := (p1; …; p_n) := (∑_jⁿ=1np _j⁻¹)⁻¹. Assume further that 0 < p ≤ 1 and that for the Hölder combinations of p2 to pn and p1 to pn−1, we have 1 ≤ (p2; …; p_n), (p1; …; p_n−1) < ∞. Then there exists a constant C > 0 such that for every (Formula presented) with ∥f⁽ⁿ⁾∥ _∞ < ∞ we have ∥T_f[n] : S_p1 ×⋯ × S_pn → S_p∥ ≤ (Formula presented). Here S_q is the Schatten–von Neumann class, Ḃ_p,q^s the homogeneous Besov space and Tf[n] is the multilinear Schur multiplier of the nth order divided difference function. In particular, our result holds for p = 1 and any 1 ≤ p1,…,p_n < ∞ with p = (p1; …; p_n).