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

Let n ∈ N≥ 1. Let 1 ≤ p₁, …, pₙ < ∞ and set the Hölder combination p := (p₁; … ; pₙ) := ( ∑ⱼ₌₁ⁿ pⱼ⁻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, \[ 1 ≤ (p_2; … ; p_n), (p_1; … ; pₙ₋₁) < ∞. \] Then there exists a constant $C> 0$ such that for every f ∈ Cⁿ(R) ∩ Ḃp/1-p, pn-1 + 1/p with f⁽ⁿ⁾ _∞ < ∞ we have \[ T_{f[n]}: Sp_1 × … × Sp_n → S_p ≤ C ( f⁽ⁿ⁾ _∞ + f _{Ḃp/1-p, pn-1 + 1/p}). \] Here Sq is the Schatten von Neumann class, Ḃp,qˢ the homogeneous Besov space, and T_f[n] is the multilinear Schur multiplier of the n-th order divided difference function. In particular, our result holds for $p=1$ and any 1 ≤ p₁, …, pₙ < ∞ with p = (p₁; …; pₙ).