{"id":496,"date":"2022-05-01T11:29:49","date_gmt":"2022-05-01T18:29:49","guid":{"rendered":"\/wp\/hades\/?p=496"},"modified":"2022-05-01T11:29:49","modified_gmt":"2022-05-01T18:29:49","slug":"implicitly-oscillatory-multilinear-integrals","status":"publish","type":"post","link":"https:\/\/wp.math.berkeley.edu\/hades\/2022\/05\/01\/implicitly-oscillatory-multilinear-integrals\/","title":{"rendered":"Implicitly Oscillatory Multilinear Integrals"},"content":{"rendered":"\n\t\t\t\t\n<p>The HADES seminar on Tuesday, <strong>May 3rd<\/strong>\u00a0will be at\u00a0<strong>3:30 pm<\/strong>\u00a0in\u00a0<strong>Room 740<\/strong>.<\/p>\n\n\n\n<p><strong>Speaker:<\/strong> Michael Christ<\/p>\n\n\n\n<p><strong>Abstract:<\/strong> An archetypal (bilinear) oscillatory integral inequality states that $$ \\Big| \\iint_{\\mathbb{R}^d\\times\\mathbb{R}^d} f(x)\\,g(y)\\,e^{i\\lambda\\phi(x,y)}\\,\\eta(x,y)\\,dx\\,dy\\Big|\\le C|\\lambda|^{-\\gamma} \\|{f}\\|_{L^2}\\|{g}\\|_{L^2}$$ where $\\lambda\\in\\mathbb{R}$ is a large parameter, $\\phi$ is a smooth real-valued phase function which is nondegenerate in a suitable sense, $f,g$ are arbitrary $L^2$ functions, $\\eta$ is a\u00a0 smooth compactly supported cutoff function, and $\\gamma&gt;0$ and $C&lt;\\infty$ depend on $\\phi$ but not on $f,g,\\lambda$. Its main features are the decaying factor $|\\lambda|^{-\\gamma}$,\u00a0the absence of any smoothness hypothesis on the measurable factors $f,g$, and the interplay between the structure of $\\phi$ and the product structure of $f(x)\\,g(y)$. If $\\phi$ is nonconstant then $e^{i\\lambda\\phi}$ oscillates rapidly, creating cancellation that potentially results in smallness of the integral. <\/p>\n\n\n\n<p>Implicitly oscillatory integrals involve no overtly oscillatory factor $e^{i\\lambda\\phi}$; instead, the measurable factors $f_j$ are themselves assumed to be oscillatory, but in a less structured way. A multilinear integral of this type takes the form \\[ \\int_{\\mathbb{R}^2} \\prod_{j=1}^N (f_j\\circ\\varphi_j)(x)\\,\\eta(x)\\,dx\\] where $\\varphi_j:\\mathbb{R}^2\\to\\mathbb{R}^1$ are smooth submersions, and the functions $f_j$ are merely measurable. The desired upper bound is expressed in terms of strictly negative order Sobolev norms of these functions. Thus the functions $f_j$ are rapidly oscillatory in the sense that they consist mainly of high frequency components.<\/p>\n\n\n\n<p>I will give an introduction to recent (and ongoing) work on this topic, and on associatedsublevel set inequalities. <br><\/p>\n\t\t","protected":false},"excerpt":{"rendered":"<p>The HADES seminar on Tuesday, May 3rd\u00a0will be at\u00a03:30 pm\u00a0in\u00a0Room 740. Speaker: Michael Christ Abstract: An archetypal (bilinear) oscillatory integral inequality states that $$ \\Big| \\iint_{\\mathbb{R}^d\\times\\mathbb{R}^d} f(x)\\,g(y)\\,e^{i\\lambda\\phi(x,y)}\\,\\eta(x,y)\\,dx\\,dy\\Big|\\le C|\\lambda|^{-\\gamma} \\|{f}\\|_{L^2}\\|{g}\\|_{L^2}$$ where $\\lambda\\in\\mathbb{R}$ is a large parameter, $\\phi$ is a smooth real-valued phase function which is nondegenerate in a suitable sense, $f,g$ are arbitrary $L^2$ functions, $\\eta$ [&hellip;]<\/p>\n","protected":false},"author":90,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-496","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/496","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/users\/90"}],"replies":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/comments?post=496"}],"version-history":[{"count":0,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/496\/revisions"}],"wp:attachment":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/media?parent=496"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/categories?post=496"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/tags?post=496"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}