Higher order perturbation estimates in quasi-Banach Schatten spaces through wavelets
M.P.T. Caspers (TU Delft - Analysis)
E. Huisman (TU Delft - Applied Sciences, TU Delft - Electrical Engineering, Mathematics and Computer Science)
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Abstract
Let n ∈ ℕ≥1. Let 1 ≤ p1,…,pn < ∞ and set the Hölder combination p := (p1; …; pn) := (∑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; …; pn), (p1; …; pn−1) < ∞. Then there exists a constant C > 0 such that for every (Formula presented) with ∥f(n)∥ ∞ < ∞ we have ∥Tf[n] : Sp1 ×⋯ × Spn → Sp∥ ≤ (Formula presented). Here Sq is the Schatten–von Neumann class, Ḃp,qs 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,…,pn < ∞ with p = (p1; …; pn).