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 w the MathML source">x₁,…,x_nden">de">. Clearly, P has a unique representation in the form and the m-form satisfies 160e154179b08c98b0b41709c92" title="Click to view the MathML source">L_P(x,…,x)=P(x)den">de"> for every w the MathML source">x∈Cⁿden">de">. We show that, although w the MathML source">L_Pden">de"> in general is non-symmetric, for a large class of reasonable norms w 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">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">den">de"> on w the MathML source">Cⁿden">de"> the norm of w the MathML source">L_Pden">de"> on w 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">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">den">de"> up to a logarithmic term w the MathML source">(clog⁡n)^m2den">de"> can be estimated by the norm of P   on w 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">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">den">de">; here w the MathML source">c≥1den">de"> denotes a universal constant. Moreover, for the w the MathML source">ℓ_pden">de">-norms w 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">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">den">de">, w the MathML source">1≤p<2den">de"> the logarithmic term in the number n of variables is even superfluous.