{"title":"On the best constants of Schur multipliers of second order divided difference functions.","authors":"Martijn Caspers, Jesse Reimann","doi":"10.1007/s00208-025-03111-y","DOIUrl":null,"url":null,"abstract":"<p><p>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 <math><mrow><mi>f</mi> <mo>∈</mo> <msup><mi>C</mi> <mn>2</mn></msup> <mrow><mo>(</mo> <mi>R</mi> <mo>)</mo></mrow> </mrow> </math> and <math><mrow><mn>1</mn> <mo><</mo> <mi>p</mi> <mo>,</mo> <msub><mi>p</mi> <mn>1</mn></msub> <mo>,</mo> <msub><mi>p</mi> <mn>2</mn></msub> <mo><</mo> <mi>∞</mi></mrow> </math> with <math> <mrow><mfrac><mn>1</mn> <mi>p</mi></mfrac> <mo>=</mo> <mfrac><mn>1</mn> <msub><mi>p</mi> <mn>1</mn></msub> </mfrac> <mo>+</mo> <mfrac><mn>1</mn> <msub><mi>p</mi> <mn>2</mn></msub> </mfrac> </mrow> </math> we have <dispformula> <math> <mrow> <mtable> <mtr> <mtd> <mrow><mrow><mo>‖</mo></mrow> <msub><mi>M</mi> <msup><mi>f</mi> <mrow><mo>[</mo> <mn>2</mn> <mo>]</mo></mrow> </msup> </msub> <mo>:</mo> <msub><mi>S</mi> <msub><mi>p</mi> <mn>1</mn></msub> </msub> <mo>×</mo> <msub><mi>S</mi> <msub><mi>p</mi> <mn>2</mn></msub> </msub> <mo>→</mo> <msub><mi>S</mi> <mi>p</mi></msub> <mrow><mo>‖</mo> <mo>≲</mo> <mo>‖</mo></mrow> <msup><mi>f</mi> <mrow><mo>'</mo> <mo>'</mo></mrow> </msup> <msub><mrow><mo>‖</mo></mrow> <mi>∞</mi></msub> <mi>D</mi> <mrow><mo>(</mo> <mi>p</mi> <mo>,</mo> <msub><mi>p</mi> <mn>1</mn></msub> <mo>,</mo> <msub><mi>p</mi> <mn>2</mn></msub> <mo>)</mo></mrow> <mo>,</mo></mrow> </mtd> </mtr> </mtable> </mrow> </math> </dispformula> where the constant <math><mrow><mi>D</mi> <mo>(</mo> <mi>p</mi> <mo>,</mo> <msub><mi>p</mi> <mn>1</mn></msub> <mo>,</mo> <msub><mi>p</mi> <mn>2</mn></msub> <mo>)</mo></mrow> </math> is specified in Theorem 7.1 and <math><mrow><mi>D</mi> <mrow><mo>(</mo> <mi>p</mi> <mo>,</mo> <mn>2</mn> <mi>p</mi> <mo>,</mo> <mn>2</mn> <mi>p</mi> <mo>)</mo></mrow> <mo>≈</mo> <msup><mi>p</mi> <mn>4</mn></msup> <msup><mi>p</mi> <mo>∗</mo></msup> </mrow> </math> with <math><msup><mi>p</mi> <mo>∗</mo></msup> </math> the Hölder conjugate of <i>p</i>. We further show that for <math><mrow><mi>f</mi> <mo>(</mo> <mi>λ</mi> <mo>)</mo> <mo>=</mo> <mi>λ</mi> <mo>|</mo> <mi>λ</mi> <mo>|</mo> <mo>,</mo></mrow> </math> <math><mrow><mi>λ</mi> <mo>∈</mo> <mi>R</mi> <mo>,</mo></mrow> </math> for every <math><mrow><mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi></mrow> </math> we have <dispformula> <math> <mrow> <mtable> <mtr> <mtd> <mrow><msup><mi>p</mi> <mn>2</mn></msup> <msup><mi>p</mi> <mo>∗</mo></msup> <mo>≲</mo> <mrow><mo>‖</mo> <msub><mi>M</mi> <msup><mi>f</mi> <mrow><mo>[</mo> <mn>2</mn> <mo>]</mo></mrow> </msup> </msub> <mo>:</mo> <msub><mi>S</mi> <mrow><mn>2</mn> <mi>p</mi></mrow> </msub> <mo>×</mo> <msub><mi>S</mi> <mrow><mn>2</mn> <mi>p</mi></mrow> </msub> <mo>→</mo> <msub><mi>S</mi> <mi>p</mi></msub> <mo>‖</mo></mrow> <mo>.</mo></mrow> </mtd> </mtr> </mtable> </mrow> </math> </dispformula> Here <math><msup><mi>f</mi> <mrow><mo>[</mo> <mn>2</mn> <mo>]</mo></mrow> </msup> </math> is the second order divided difference function of <i>f</i> with <math><msub><mi>M</mi> <msup><mi>f</mi> <mrow><mo>[</mo> <mn>2</mn> <mo>]</mo></mrow> </msup> </msub> </math> the associated Schur multiplier. In particular it follows that our estimate <i>D</i>(<i>p</i>, 2<i>p</i>, 2<i>p</i>) is optimal for <math><mrow><mi>p</mi> <mo>↘</mo> <mn>1</mn> <mo>.</mo></mrow></math></p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"392 1","pages":"1119-1166"},"PeriodicalIF":1.3000,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11971180/pdf/","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematische Annalen","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00208-025-03111-y","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2025/3/3 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
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 and with we have where the constant is specified in Theorem 7.1 and with the Hölder conjugate of p. We further show that for for every we have Here is the second order divided difference function of f with the associated Schur multiplier. In particular it follows that our estimate D(p, 2p, 2p) is optimal for
我们给出了二阶微分函数双线性Schur乘子的有界性的一个新的证明,这个证明是先前由Potapov、Skripka和Sukochev在证明Koplienko关于高阶谱移函数存在性的猜想中得到的。我们的证明是基于最近的方法,涉及双线性迁移和Hörmander-Mikhlin-Schur乘数定理。我们的方法提供了二阶可分差分函数的双线性舒尔乘子的已知渐近界的显著锐化。此外,我们给出了这些双线性舒尔乘子的一个新的下界,再次对Coine、Le Merdy、Potapov、Sukochev和Tomskova等人得到的最著名的下界作了根本的改进。更准确地说,我们证明了f∈C 2 (R)和1 p, p 1, p 2∞1 p = 1 p 1 + 1 p 2我们已经为M f [2]: S p 1×S p 2→S p为≲为f ' '为∞D (1 p, p, p 2 ) , 常数D (1 p, p, p 2)指定在定理7.1和D (p 2 p 2 p)≈p 4 p∗∗p的持有人共轭。我们进一步表明,f(λ)=λ|λ|,λ∈R,每1 p∞p 2 p∗≲为M f [2]: 2 p×年代2 p→S p为。这里f b[2]是f与M的二阶差分函数f b[2]是相关的舒尔乘子。特别地,我们的估计D(p, 2p, 2p)对于p ` ` 1是最优的。
期刊介绍:
Begründet 1868 durch Alfred Clebsch und Carl Neumann. Fortgeführt durch Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück und Nigel Hitchin.
The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. It was continued by Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguigon, Wolfgang Lück and Nigel Hitchin.
Since 1868 the name Mathematische Annalen stands for a long tradition and high quality in the publication of mathematical research articles. Mathematische Annalen is designed not as a specialized journal but covers a wide spectrum of modern mathematics.