Non-symmetric polarization

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• 作者：Andreas Defant ; ^{andreas.defant@uni-oldenburg.de" class="auth_mail" title="E-mail the corresponding author} ; Sunke Schlü ; ters ^{sunke.schlueters@uni-oldenburg.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Polarization ; Polynomials ; Multilinear forms ; Schur multipliers
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：445
• 期：2
• 页码：1291-1299
• 全文大小：306 K

文摘

Let P be an m-homogeneous polynomial in n  -complex variables class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si1.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=d14882934c16a217cb876931f2ad3c9d" title="Click to view the MathML source">x₁,…,x_nclass="mathContainer hidden">class="mathCode">$x_1,\dots ,x_n$. Clearly, P has a unique representation in the form and the m-form satisfies class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si4.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=c495f160e154179b08c98b0b41709c92" title="Click to view the MathML source">L_P(x,…,x)=P(x)class="mathContainer hidden">class="mathCode">$L_P(x,\dots ,x)=P(x)$ for every class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si21.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=51e24c7c4f844e450d0fc0c78a333a7a" title="Click to view the MathML source">x∈Cⁿclass="mathContainer hidden">class="mathCode">$x\in {\mathbb{C}}^n$. We show that, although class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si6.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=e4cb0fb37682b5107f09ff783edacefc" title="Click to view the MathML source">L_Pclass="mathContainer hidden">class="mathCode">$L_P$ in general is non-symmetric, for a large class of reasonable norms class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si212.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=e23746b6c197461583f071dafac35731">class="imgLazyJSB inlineImage" height="17" width="26" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16302414-si212.gif">class="mathContainer hidden">class="mathCode">$\Vert \cdot \Vert$ on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si19.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=847d41fe858350f8184d8533ad8f9dd1" title="Click to view the MathML source">Cⁿclass="mathContainer hidden">class="mathCode">${\mathbb{C}}^n$ the norm of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si6.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=e4cb0fb37682b5107f09ff783edacefc" title="Click to view the MathML source">L_Pclass="mathContainer hidden">class="mathCode">$L_P$ on class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si9.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=e26457f8dec697251cee531644bf3870">class="imgLazyJSB inlineImage" height="17" width="79" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16302414-si9.gif">class="mathContainer hidden">class="mathCode">$({\mathbb{C}}^n,\Vert \cdot {\Vert )}^m$ up to a logarithmic term class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si10.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=22c4d36893e9c3576ac3897fd6e65ce7" title="Click to view the MathML source">(clog⁡n)^m2class="mathContainer hidden">class="mathCode">${(c\log n)}^{m^2}$ can be estimated by the norm of P   on class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si11.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=994c7cebb58fe9ae6d866fabaf63e5b2">class="imgLazyJSB inlineImage" height="17" width="66" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16302414-si11.gif">class="mathContainer hidden">class="mathCode">$({\mathbb{C}}^n,\Vert \cdot \Vert )$; here class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si12.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=930e6572fc7b46b1f8431e45a204f058" title="Click to view the MathML source">c≥1class="mathContainer hidden">class="mathCode">$c\ge 1$ denotes a universal constant. Moreover, for the class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si13.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=dd37c333e43863f7b1bfd6a9e351210a" title="Click to view the MathML source">ℓ_pclass="mathContainer hidden">class="mathCode">${\ell }_p$-norms class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si14.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=4bda895b2e18297f29ce4d865eeedf42">class="imgLazyJSB inlineImage" height="17" width="34" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16302414-si14.gif">class="mathContainer hidden">class="mathCode">${\Vert \cdot \Vert }_p$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si15.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=13ce8527e3073507043352050fe22260" title="Click to view the MathML source">1≤p<2class="mathContainer hidden">class="mathCode">$1\le p<2$ the logarithmic term in the number n of variables is even superfluous.