On the best constants of Schur multipliers of second order divided difference functions
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arXiv
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| Natura: | Preprint |
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2024
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| author | Caspers, Martijn Reimann, Jesse |
| author_facet | Caspers, Martijn Reimann, Jesse |
| contents | 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 $p \searrow 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_00464 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the best constants of Schur multipliers of second order divided difference functions Caspers, Martijn Reimann, Jesse Classical Analysis and ODEs Functional Analysis Operator Algebras 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 $p \searrow 1$. |
| title | On the best constants of Schur multipliers of second order divided difference functions |
| topic | Classical Analysis and ODEs Functional Analysis Operator Algebras |
| url | https://arxiv.org/abs/2405.00464 |