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Our main result is a Hörmander-Mihlin multiplier theorem for finite-dimensional cocycles with optimal smoothness condition. We also find Littlewood-Paley type" /> <meta name="twitter:image" content="https://0.academia-photos.com/146358351/78569153/67122316/s200_marius.junge.png" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/54047195/Aspects_of_Calder_on_Zygmund_theory_for_von_Neumann_algebras_I" /> <meta property="og:title" content="Aspects of Calder 'on-Zygmund theory for von Neumann algebras I" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="We investigate Fourier multipliers on the compact dual of arbitrary discrete groups. Our main result is a Hörmander-Mihlin multiplier theorem for finite-dimensional cocycles with optimal smoothness condition. We also find Littlewood-Paley type" /> <meta property="article:author" content="https://independent.academia.edu/JungeM" /> <meta name="description" content="We investigate Fourier multipliers on the compact dual of arbitrary discrete groups. Our main result is a Hörmander-Mihlin multiplier theorem for finite-dimensional cocycles with optimal smoothness condition. We also find Littlewood-Paley type" /> <title>(PDF) Aspects of Calder 'on-Zygmund theory for von Neumann algebras I | Marius Junge - Academia.edu</title> <link rel="canonical" href="https://www.academia.edu/54047195/Aspects_of_Calder_on_Zygmund_theory_for_von_Neumann_algebras_I" /> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "single_work", 'action': "show", 'controller_action': 'single_work#show', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two cases. // ab_test_bucket should be of the form <ab_test_name>:<bucket> 'ab_test_bucket': null, }) </script> <script> var $controller_name = 'single_work'; var $action_name = "show"; var $rails_env = 'production'; var $app_rev = 'd0d4cf753dac7d44fa55ec8140bebbf4477d3bc9'; var $domain = 'academia.edu'; var $app_host = "academia.edu"; var $asset_host = "academia-assets.com"; var $start_time = new Date().getTime(); var $recaptcha_key = "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB"; var $recaptcha_invisible_key = "6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj"; var $disableClientRecordHit = false; </script> <script> window.require = { config: function() { return function() {} } } </script> <script> window.Aedu = window.Aedu || {}; window.Aedu.hit_data = null; window.Aedu.serverRenderTime = new Date(1733261255000); window.Aedu.timeDifference = new Date().getTime() - 1733261255000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"We investigate Fourier multipliers on the compact dual of arbitrary discrete groups. Our main result is a Hörmander-Mihlin multiplier theorem for finite-dimensional cocycles with optimal smoothness condition. We also find Littlewood-Paley type inequalities in group von Neumann algebras, prove $L_p$ estimates for noncommutative Riesz transforms and characterize $L_\\infty \\to \\mathrm{BMO}$ boundedness for radial Fourier multipliers. The key novelties of our approach are to exploit group cocycles and cross products in Fourier multiplier theory in conjunction with BMO spaces associated to semigroups of operators and a noncommutative generalization of Calderón-Zygmund theory.","author":[{"@context":"https://schema.org","@type":"Person","name":"Marius Junge"}],"contributor":[],"dateCreated":"2021-09-29","datePublished":"2010-01-01","headline":"Aspects of Calder 'on-Zygmund theory for von Neumann algebras I","image":"https://attachments.academia-assets.com/70600146/thumbnails/1.jpg","inLanguage":"en","keywords":["Harmonic Analysis","Geometric group theory","von Neumann algebra","Free 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window.loswp.shouldShowBulkDownload = true; window.loswp.showSignupCaptcha = false window.loswp.willEdgeCache = false; window.loswp.work = {"work":{"id":54047195,"created_at":"2021-09-29T14:40:19.410-07:00","from_world_paper_id":174742049,"updated_at":"2024-08-03T12:05:02.159-07:00","_data":{"abstract":"We investigate Fourier multipliers on the compact dual of arbitrary discrete groups. Our main result is a Hörmander-Mihlin multiplier theorem for finite-dimensional cocycles with optimal smoothness condition. We also find Littlewood-Paley type inequalities in group von Neumann algebras, prove $L_p$ estimates for noncommutative Riesz transforms and characterize $L_\\infty \\to \\mathrm{BMO}$ boundedness for radial Fourier multipliers. The key novelties of our approach are to exploit group cocycles and cross products in Fourier multiplier theory in conjunction with BMO spaces associated to semigroups of operators and a noncommutative generalization of Calderón-Zygmund theory.","publication_date":"2010,,"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Aspects of Calder 'on-Zygmund theory for von Neumann algebras I","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [146358351]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "control"; window.loswp.useOptimizedScribd4genScript = false; window.loswp.appleClientId = 'edu.academia.applesignon';</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":70600146,"attachmentType":"pdf"}"><img alt="First page of “Aspects of Calder 'on-Zygmund theory for von Neumann algebras I”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/70600146/mini_magick20210929-31158-jr7sj5.png?1632952279" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/assets/single_work_splash/adobe.icon-574afd46eb6b03a77a153a647fb47e30546f9215c0ee6a25df597a779717f9ef.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">Aspects of Calder 'on-Zygmund theory for von Neumann algebras I</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="146358351" href="https://independent.academia.edu/JungeM"><img alt="Profile image of Marius Junge" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/146358351/78569153/67122316/s65_marius.junge.png" />Marius Junge</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2010</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">56 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 54047195; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=54047195"; const getWorkViews = async (workId) => { const response = await fetch(worksViewsPath); if (!response.ok) { throw new Error('Failed to load work views'); } const data = await response.json(); return data.views[workId]; }; // Get the view count for the work - we send this immediately rather than waiting for // the DOM to load, so it can be available as soon as possible (but without holding up // the backend or other resource requests, because it's a bit expensive and not critical). const viewCount = await getWorkViews(workId); const updateViewCount = (viewCount) => { const viewCountNumber = Number(viewCount); if (!viewCountNumber) { throw new Error('Failed to parse view count'); } const commaizedViewCount = viewCountNumber.toLocaleString(); const viewCountBody = document.getElementById('work-metadata-view-count'); if (viewCountBody) { viewCountBody.textContent = `${commaizedViewCount} views`; } else { throw new Error('Failed to find work views element'); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">We investigate Fourier multipliers on the compact dual of arbitrary discrete groups. Our main result is a Hörmander-Mihlin multiplier theorem for finite-dimensional cocycles with optimal smoothness condition. We also find Littlewood-Paley type inequalities in group von Neumann algebras, prove $L_p$ estimates for noncommutative Riesz transforms and characterize $L_\infty \to \mathrm{BMO}$ boundedness for radial Fourier multipliers. The key novelties of our approach are to exploit group cocycles and cross products in Fourier multiplier theory in conjunction with BMO spaces associated to semigroups of operators and a noncommutative generalization of Calderón-Zygmund theory.</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":70600146,"attachmentType":"pdf","workUrl":"https://www.academia.edu/54047195/Aspects_of_Calder_on_Zygmund_theory_for_von_Neumann_algebras_I"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":70600146,"attachmentType":"pdf","workUrl":"https://www.academia.edu/54047195/Aspects_of_Calder_on_Zygmund_theory_for_von_Neumann_algebras_I"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download 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class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="55648107" href="https://independent.academia.edu/MariusJunge">Marius Junge</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Geometric and Functional Analysis, 2014</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Smooth Fourier multipliers on group von Neumann algebras","attachmentId":49882749,"attachmentType":"pdf","work_url":"https://www.academia.edu/29436596/Smooth_Fourier_multipliers_on_group_von_Neumann_algebras","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/29436596/Smooth_Fourier_multipliers_on_group_von_Neumann_algebras"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="1" data-entity-id="54535373" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/54535373/On_the_vector_Fourier_multipliers_for_compact_groups">On the vector Fourier multipliers for compact groups</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="172262998" href="https://independent.academia.edu/YaoganMensah">Yaogan Mensah</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Open Journal of Mathematical Sciences</p><div class="ds-related-work--ctas"><button 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data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/54047204/Noncommutative_Riesz_transforms_dimension_free_bounds_and_Fourier_multipliers">Noncommutative Riesz transforms – dimension free bounds and Fourier multipliers</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="146358351" href="https://independent.academia.edu/JungeM">Marius Junge</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of the European Mathematical Society</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Noncommutative Riesz transforms – dimension free bounds and Fourier 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The combination of these ingredients yiel...</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Smooth Fourier multipliers in group algebras via Sobolev dimension","attachmentId":83901654,"attachmentType":"pdf","work_url":"https://www.academia.edu/72846672/Smooth_Fourier_multipliers_in_group_algebras_via_Sobolev_dimension","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/72846672/Smooth_Fourier_multipliers_in_group_algebras_via_Sobolev_dimension"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="54535363" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/54535363/Completely_Bounded_Fourier_Multipliers_Over_Compact_Groups">Completely Bounded Fourier Multipliers Over Compact Groups</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="172262998" href="https://independent.academia.edu/YaoganMensah">Yaogan Mensah</a></div><p class="ds-related-work--abstract ds2-5-body-sm">Vector version of Fourier multipliers over compact non necessary abelian groups are defined. 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For 1 ≤ p &lt; ∞, denote by Ap(G) the space of integrable functions on G whose Fourier transforms belong to lp(Γ). We investigate several problems related to multipliers from Ap(G) to Aq(G). In particular, we prove that (Ap, Ap) ⊊ (Aq, Aq). 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