Lipschitz estimates in quasi-Banach Schatten ideals

Estimates for Schur Multipliers and Double Operator Integrals -- A Wavelet Approach

“…Before we proceed, let us note that for a function k ∈ Ḃs p,q (R, L ∞ (R)), it need not be the case that the decomposition k = j∈Z k j holds (compare to the scalarvalued failure of such a decomposition, as detailed in [21, Chapter 3, Proposition 4]). Despite this, it follows from an obvious modification of [19,Lemma 4.1.4] that if k ∈ L ∞ (R 2 ), then there exists a function c ∈ L ∞ (R), such that k(t, s) = c(t) + j∈Z (k j (t, s) − k j (t, 0)) , for all t, s ∈ R 2 , and such that c ∞ k ∞ + |k| Ḃs p,q (R,L∞(R)) . Finally, we may state our strengthening of [5,Theorem 9.2].…”

“…However, there are some clear examples, such as the early work of Peng [22], who used wavelet bases to find Schatten-von Neumann class norm estimates for certain integral operators. Only recently, these techniques were adapted in order to prove new Lipschitz estimates in Schatten-von Neumann ideals [19] We mark three contributions to the theory of DOIs. The first contribution is to extend Schatten-von Neumann class and Schur multiplier estimates to DOIs having symbol with non-compact support.…”

Operator θ$\theta$‐Hölder functions with respect to ∥·∥p$\Vert \cdot \Vert _p$, 0<p⩽∞$0&lt; p\leqslant \infty$

“…However, it has been shown recently by Potapov and Sukochev [71] that every Lipschitz function is necessarily operator‐Lipschitz in the Schatten–von Neumann ideal $$S_p$$ (or non‐commutative $$L_p$$‐space) for any $$1&lt;p&lt;\infty$$ (see [19] for the best constant for operator Lipschitz functions). A class of operator‐Lipschitz functions for Schatten–von Neumann ideal $$S_p$$ for $$p&lt;1$$ has been identified in [60].…”