The problem of ideals of H^∞: Beyond the exponent 3/2

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• 作者：Sergei Treil
• 关键词：Corona Theorem ; Problem of ideals ; Carleson measures ; Hermitian vector bundles ; Bellman function
• 刊名：Journal of Functional Analysis
• 出版年：2007
• 出版时间：1 December 2007
• 年：2007
• 卷：253
• 期：1
• 页码：220-240
• 全文大小：208 K

文摘

The paper deals with the problem of ideals of href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WJJ-4PPNMR3-3&_mathId=mml2&_user=10&_cdi=6880&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=51c658236b3fc4d439708b24bc67a920"" title=""Click to view the MathML source"">H^∞: describe increasing functions href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WJJ-4PPNMR3-3&_mathId=mml3&_user=10&_cdi=6880&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=14887b8e70f8a5bfda054cebd62af33c"" title=""Click to view the MathML source"">φhttp://www.sciencedirect.com/scidirimg/entities/2a7e.gif"" alt=""greater-or-equal, slanted"" border=0>0 such that for all bounded analytic functions l4"">href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WJJ-4PPNMR3-3&_mathId=mml4&_user=10&_cdi=6880&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=3b6c49ecda12c8a4d1a56d7c6a0a20e7"" title=""Click to view the MathML source"">f₁,f₂,…,f_n,τ in the unit disc href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WJJ-4PPNMR3-3&_mathId=mml5&_user=10&_cdi=6880&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=c47a5f6d56805038895924ddc8642604"">http://www.sciencedirect.com/cache/MiamiImageURL/B6WJJ-4PPNMR3-3-J/0?wchp=dGLzVzz-zSkzV"" alt=""Click to view the MathML source"" align=""absbottom"" border=""0"" height=12 width=14> the condition

width=""100 % "" border=""0"">

width=""95 % "">href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WJJ-4PPNMR3-3&_mathId=mml6&_user=10&_cdi=6880&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=2cd9e664c045220b8454a6cbe008343f"">http://www.sciencedirect.com/cache/MiamiImageURL/B6WJJ-4PPNMR3-3-K/0?wchp=dGLzVzz-zSkzV"" alt=""Click to view the MathML source"" align=""absbottom"" border=""0"" height=33 width=259>

implies that τ belong to the ideal generated by href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WJJ-4PPNMR3-3&_mathId=mml7&_user=10&_cdi=6880&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=dab49330a6177390a1d00353bb938cab"" title=""Click to view the MathML source"">f₁,f₂,…,f_n, i.e. that there exist bounded analytic functions href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WJJ-4PPNMR3-3&_mathId=mml8&_user=10&_cdi=6880&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=0a4dbf5a47e080a7270a071d07b5f3cc"" title=""Click to view the MathML source"">g₁,g₂,…,g_n such that href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WJJ-4PPNMR3-3&_mathId=mml9&_user=10&_cdi=6880&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=7657ed4d93ce544448b927d6a69ba3be"">http://www.sciencedirect.com/cache/MiamiImageURL/B6WJJ-4PPNMR3-3-P/0?wchp=dGLzVzz-zSkzV"" alt=""Click to view the MathML source"" align=""absbottom"" border=""0"" height=18 width=101>.

It was proved earlier by the author that the function href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WJJ-4PPNMR3-3&_mathId=mml10&_user=10&_cdi=6880&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=078a70555d09b07567813da5a98511e0"" title=""Click to view the MathML source"">φ(s)=s² does not satisfy this condition. The strongest known positive result in this direction due to J. Pau states that the function href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WJJ-4PPNMR3-3&_mathId=mml11&_user=10&_cdi=6880&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=460d9f06df679392842412e0589dd43c"" title=""Click to view the MathML source"">φ(s)=s²/((lns⁻¹)^3/2lnlns⁻¹) works. However, there was always a suspicion that the critical exponent at href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WJJ-4PPNMR3-3&_mathId=mml12&_user=10&_cdi=6880&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=f0a4d7907a45ca05dea9cb1375f88214"" title=""Click to view the MathML source"">lns⁻¹ is 1 and not 3/2.

This suspicion turned out (at least partially) to be true, 3/2 indeed is not the critical exponent. The main result of the paper is that one can take for φ any function of form href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WJJ-4PPNMR3-3&_mathId=mml13&_user=10&_cdi=6880&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=c3b458eb5bf430c3674a24668c1304d2"" title=""Click to view the MathML source"">φ(s)=s²ψ(lns⁻²), where href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WJJ-4PPNMR3-3&_mathId=mml14&_user=10&_cdi=6880&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=dbb1e1486b1c20ac0f809c799a171b9a"">http://www.sciencedirect.com/cache/MiamiImageURL/B6WJJ-4PPNMR3-3-7/0?wchp=dGLzVzz-zSkzV"" alt=""Click to view the MathML source"" align=""absbottom"" border=""0"" height=16 width=90> is a bounded non-increasing function satisfying href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WJJ-4PPNMR3-3&_mathId=mml15&_user=10&_cdi=6880&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=5bf2f83be0dc5c75591cc202e28f11ee"">http://www.sciencedirect.com/cache/MiamiImageURL/B6WJJ-4PPNMR3-3-8/0?wchp=dGLzVzz-zSkzV"" alt=""Click to view the MathML source"" align=""absbottom"" border=""0"" height=20 width=117>. In particular any of the functions

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width=""95 % "">href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WJJ-4PPNMR3-3&_mathId=mml16&_user=10&_cdi=6880&_rdoc=7&_acct=C000050221&_version=1&_userid=10&md5=760e483afceeab67c3587d7f46f37851"">http://www.sciencedirect.com/cache/MiamiImageURL/B6WJJ-4PPNMR3-3-R/0?wchp=dGLzVzz-zSkzV"" alt=""Click to view the MathML source"" align=""absbottom"" border=""0"" height=41 width=509>

works.