On the best constants of Schur multipliers of second order divided difference functions
Martijn Caspers, Jesse Reimann
TL;DR
The paper provides a new, sharp proof of the boundedness of bilinear Schur multipliers associated with second-order divided differences, using bilinear transference and Hörmander-Mikhlin–Schur multiplier techniques. It decomposes f^{[2]} into linear and bilinear Toeplitz components, bounding each piece via a mix of HMS-type theorems and Calderón-Zygmund theory, yielding upper bounds with asymptotics D(p, p1, p2) that improve prior results. A key achievement is identifying precise p-behavior, showing D(p,2p,2p) ≈ p^4 p^* as p→∞ and establishing lower bounds for specific choices of f (notably f(s)=s|s|) that match these asymptotics in various limits, thereby confirming near-optimality. An extrapolation result to the Marcinkiewicz-type space M_{1,∞} is provided, together with comprehensive lower-bound results, clarifying the endpoint behavior and the range of applicability of the main estimates. Overall, the work advances the understanding of higher-order spectral shift phenomena by providing both sharper upper bounds and fundamental lower bounds for bilinear Schur multipliers on Schatten classes.
Abstract
We give a new proof of the boundedness of bilinear Schur multipliers of second order divided difference functions, as obtained earlier by Potapov, Skripka and Sukochev in their proof of Koplienko's conjecture on the existence of higher order spectral shift functions. Our proof is based on recent methods involving bilinear transference and the Hörmander-Mikhlin-Schur multiplier theorem. Our approach provides a significant sharpening of the known asymptotic bounds of bilinear Schur multipliers of second order divided difference functions. Furthermore, we give a new lower bound of these bilinear Schur multipliers, giving again a fundamental improvement on the best known bounds obtained by Coine, Le Merdy, Potapov, Sukochev and Tomskova. More precisely, we prove that for $f \in C^2(\mathbb{R})$ and $1 < p, p_1, p_2 < \infty$ with $\frac{1}{p} = \frac{1}{p_1} + \frac{1}{p_2}$ we have \[ \Vert M_{f^{[2]}}: S_{p_1} \times S_{p_2} \rightarrow S_p \Vert \lesssim \Vert f'' \Vert_\infty D(p, p_1, p_2), \] where the constant $D(p, p_1, p_2)$ is specified in Theorem 7.1 and $D(p, 2p, 2p) \approx p^4 p^\ast$ with $p^\ast$ the Hölder conjugate of $p$. We further show that for $f(λ) = λ\vert λ\vert$, $λ\in \mathbb{R}$, for every $1 < p < \infty$ we have \[ p^2 p^\ast \lesssim \Vert M_{f^{[2]}}: S_{2p} \times S_{2p} \rightarrow S_p \Vert. \] Here $f^{[2]}$ is the second order divided difference function of $f$ with $M_{f^{[2]}}$ the associated Schur multiplier. In particular it follows that our estimate $D(p, 2p, 2p)$ is optimal for $p \searrow 1$.