{"id":30,"date":"2009-06-11T01:25:23","date_gmt":"2009-06-11T05:25:23","guid":{"rendered":"http:\/\/blogs.williams.edu\/Morgan\/?p=30"},"modified":"2012-07-06T05:47:51","modified_gmt":"2012-07-06T10:47:51","slug":"sobolev-type-inequality","status":"publish","type":"post","link":"https:\/\/sites.williams.edu\/Morgan\/2009\/06\/11\/sobolev-type-inequality\/","title":{"rendered":"Log-Sobolev Inequality"},"content":{"rendered":"<p>My 2009 <a href=\"http:\/\/math.williams.edu\/small\">Williams College NSF &#8220;SMALL&#8221;<\/a> undergraduate research <a href=\"http:\/\/arxiv.org\/abs\/1012.0450\">Geometry Group<\/a> has the following inequality for any <img src='https:\/\/s0.wp.com\/latex.php?latex=C%5E1&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='C^1' title='C^1' class='latex' \/> function on the unit interval and for any p \u2265 1:<\/p>\n<p><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cleft%28%5Cint_0%5E1%7Bf%5E%7B%5Cfrac%7Bp%2B1%7D%7Bp%7D%7D%7D%5Cright%29%5E%5Cfrac%7Bp%7D%7Bp%2B1%7D%5Cle%5Cint_0%5E1%7B%5Cleft%28f%5E2%2Bf%27%5E2%2F%5Cpi%5E2%5Cright%29%5E%7B1%2F2%7D%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\left(\\int_0^1{f^{\\frac{p+1}{p}}}\\right)^\\frac{p}{p+1}\\le\\int_0^1{\\left(f^2+f&#039;^2\/\\pi^2\\right)^{1\/2}}' title='\\left(\\int_0^1{f^{\\frac{p+1}{p}}}\\right)^\\frac{p}{p+1}\\le\\int_0^1{\\left(f^2+f&#039;^2\/\\pi^2\\right)^{1\/2}}' class='latex' \/><\/p>\n<p>with equality for constant functions and if p&gt;1 only for constant functions. They conjecture that these results still hold if <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cpi%5E2&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\pi^2' title='\\pi^2' class='latex' \/> on the right-hand side is replaced by <img src='https:\/\/s0.wp.com\/latex.php?latex=p%5Cpi%5E2&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='p\\pi^2' title='p\\pi^2' class='latex' \/> (sharp).<\/p>\n<p>The case p=1 is standard and follows from Wirtinger&#8217;s Inequality.<\/p>\n<p>Are any inequalities like this known?<\/p>\n","protected":false},"excerpt":{"rendered":"<p>My 2009 Williams College NSF &#8220;SMALL&#8221; undergraduate research Geometry Group has the following inequality for any function on the unit interval and for any p \u2265 1: with equality for constant functions and if p&gt;1 only for constant functions. They conjecture that these results still hold if on the right-hand side is replaced by (sharp). [&hellip;]<\/p>\n","protected":false},"author":269,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[14043],"tags":[],"class_list":["post-30","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/30","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/users\/269"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/comments?post=30"}],"version-history":[{"count":9,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/30\/revisions"}],"predecessor-version":[{"id":669,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/30\/revisions\/669"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=30"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/categories?post=30"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/tags?post=30"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}