[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/interpolation-inequality-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/interpolation-inequality-wikipedia\/","headline":"Interpolation inequality – Wikipedia","name":"Interpolation inequality – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 In the field of mathematical analysis, an interpolation inequality is an inequality of","datePublished":"2021-10-01","dateModified":"2021-10-01","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c55672b4d202e284add47db147097aad17e43cb9","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c55672b4d202e284add47db147097aad17e43cb9","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/interpolation-inequality-wikipedia\/","wordCount":6284,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In the field of mathematical analysis, an interpolation inequality is an inequality of the form       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u2016u0\u20160\u2264C\u2016u1\u20161\u03b11\u2016u2\u20162\u03b12\u2026\u2016un\u2016n\u03b1n,n\u22652,{displaystyle |u_{0}|_{0}leq C|u_{1}|_{1}^{alpha _{1}}|u_{2}|_{2}^{alpha _{2}}dots |u_{n}|_{n}^{alpha _{n}},quad ngeq 2,}where for 0\u2264k\u2264n{displaystyle 0leq kleq n},  uk{displaystyle u_{k}} is an element of some particular vector space Xk{displaystyle X_{k}} equipped with norm        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u2016\u22c5\u2016k{displaystyle |cdot |_{k}} and \u03b1k{displaystyle alpha _{k}} is some real exponent,  and C{displaystyle C} is some constant independent of u0,..,un{displaystyle u_{0},..,u_{n}}. The vector spaces concerned are usually function spaces, and many interpolation inequalities assume u0=u1=\u22ef=un{displaystyle u_{0}=u_{1}=cdots =u_{n}} and so bound the norm of an element in one space with a combination norms in other spaces, such as Ladyzhenskaya’s inequality and the Gagliardo-Nirenberg interpolation inequality, both given below. Nonetheless, some important interpolation inequalities involve distinct elements u0,..,un{displaystyle u_{0},..,u_{n}}, including H\u00f6lder’s Inequality and Young’s inequality for convolutions which are also presented below.Table of ContentsApplications[edit]Examples[edit]Examples of interpolation inequalities[edit]References[edit]Applications[edit]The main applications of interpolation inequalities lie in fields of study, such as partial differential equations, where various function spaces are used. An important example are the Sobolev spaces, consisting of functions whose weak derivatives up to some (not necessarily integer) order lie in Lp spaces for some p. There interpolation inequalities are used, roughly speaking, to bound derivatives of some order with a combination of derivatives of other orders. They can also be used to bound products, convolutions, and other combinations of functions, often with some flexibility in the choice of function space. Interpolation inequalities are fundamental to the notion of an interpolation space, such as the space Ws,p{displaystyle W^{s,p}}, which loosely speaking is composed of functions whose sth{displaystyle s^{th}} order weak derivatives lie in Lp{displaystyle L^{p}}. Interpolation inequalities are also applied when working with Besov spaces Bp,qs(\u03a9){displaystyle B_{p,q}^{s}(Omega )}, which are a generalization of the Sobolev spaces.[1] Another class of space admitting interpolation inequalities are the H\u00f6lder spaces.Examples[edit]A simple example of an interpolation inequality \u2014 one in which all the uk are the same u, but the norms \u2016\u00b7\u2016k are different \u2014 is Ladyzhenskaya’s inequality for functions u:\u00a0\u211d2\u00a0\u2192\u00a0\u211d, which states that whenever u is a compactly supported function such that both u and its gradient \u2207u are square integrable, it follows that the fourth power of u is integrable and[2]\u222bR2|u(x)|4dx\u22642\u222bR2|u(x)|2dx\u222bR2|\u2207u(x)|2dx,{displaystyle int _{mathbb {R} ^{2}}|u(x)|^{4},mathrm {d} xleq 2int _{mathbb {R} ^{2}}|u(x)|^{2},mathrm {d} xint _{mathbb {R} ^{2}}|nabla u(x)|^{2},mathrm {d} x,}i.e.\u2016u\u2016L4\u226424\u2016u\u2016L21\/2\u2016\u2207u\u2016L21\/2.{displaystyle |u|_{L^{4}}leq {sqrt[{4}]{2}},|u|_{L^{2}}^{1\/2},|nabla u|_{L^{2}}^{1\/2}.}A slightly weaker form of Ladyzhenskaya’s inequality applies in dimension 3, and Ladyzhenskaya’s inequality is actually a special case of a general result that subsumes many of the interpolation inequalities involving Sobolev spaces, the Gagliardo-Nirenberg interpolation inequality.[3]The following example, this one allowing interpolation of non-integer Sobolev spaces, is also a special case of the Gagliardo-Nirenberg interpolation inequality.[4] Denoting the L2{displaystyle L^{2}} Sobolev spaces by Hk=Wk,2{displaystyle H^{k}=W^{k,2}}, and given real numbers 1\u2264kHm{displaystyle uin H^{m}}, we have\u2016u\u2016H\u2113\u2264\u2016u\u2016Hkm\u2212\u2113m\u2212k\u2016u\u2016Hm\u2113\u2212km\u2212k{displaystyle |u|_{H^{ell }}leq |u|_{H^{k}}^{frac {m-ell }{m-k}}|u|_{H^{m}}^{frac {ell -k}{m-k}}}An example of an interpolation inequality where the elements differ is Young’s inequality for convolutions.[5] Given exponents 1\u2264p,q,r\u2264\u221e{displaystyle 1leq p,q,rleq infty }  such that 1p+1q=1+1r{displaystyle {tfrac {1}{p}}+{tfrac {1}{q}}=1+{tfrac {1}{r}}} and functions f\u2208Lp,\u00a0g\u2208Lq{displaystyle fin L^{p}, gin L^{q}}, their convolution lies in Lr{displaystyle L^{r}} and\u2016f\u2217g\u2016Lr\u2264\u2016f\u2016Lp\u2016g\u2016Lq{displaystyle |f*g|_{L^{r}}leq |f|_{L^{p}}|g|_{L^{q}}}The well known H\u00f6lder’s inequality[3] is another of this type: given 1\u2264p,q\u2264\u221e{displaystyle 1leq p,qleq infty } and functions f\u2208Lp(\u03a9),\u00a0g\u2208Lq(\u03a9){displaystyle fin L^{p}(Omega ), gin L^{q}(Omega )} with 1\/p+1\/q=1{displaystyle 1\/p+1\/q=1}, their product is in L1(\u03a9){displaystyle L^{1}(Omega )} and\u2016fg\u2016L1\u2264\u2016f\u2016Lp\u2016g\u2016Lq{displaystyle |fg|_{L^{1}}leq |f|_{L^{p}}|g|_{L^{q}}}Examples of interpolation inequalities[edit]References[edit]^ DeVore, Ronald A.; Popov, Vasil A. (1988). “Interpolation of Besov spaces”. Transactions of the American Mathematical Society. 305 (1): 397\u2013414. doi:10.1090\/S0002-9947-1988-0920166-3. ISSN\u00a00002-9947.^ Foias, C.; Manley, O.; Rosa, R.; Temam, R. (2001). Navier-Stokes Equations and Turbulence. Encyclopedia of Mathematics and its Applications. Cambridge: Cambridge University Press. doi:10.1017\/cbo9780511546754. ISBN\u00a0978-0-521-36032-6.^ a b Evans, Lawrence C. (2010). Partial differential equations (2\u00a0ed.). Providence, R.I. ISBN\u00a0978-0-8218-4974-3. OCLC\u00a0465190110.^ Br\u00e9zis, H. (2011). Functional analysis, Sobolev spaces and partial differential equations. H.. Br\u00e9zis. New York: Springer. p.\u00a0233. ISBN\u00a0978-0-387-70914-7. OCLC\u00a0695395895.^ Leoni, Giovanni (2017). A first course in Sobolev spaces (2\u00a0ed.). Providence, Rhode Island. ISBN\u00a0978-1-4704-2921-8. OCLC\u00a0976406106.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/interpolation-inequality-wikipedia\/#breadcrumbitem","name":"Interpolation inequality – Wikipedia"}}]}]