Lipschitz-type conditions on homogeneous Banach spaces of analytic functions

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• 作者：Oscar Blasco^a ; ¹ ; ^{oblasco@uv.es" class="auth_mail" title="E-mail the corresponding author} ; Georgios Stylogiannis^b ; ^{stylog@math.auth.gr" class="auth_mail" title="E-mail the corresponding author} ; ^{g.stylog@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Banach spaces ; Lipschitz-type conditions ; Approximation by partial sums
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：445
• 期：1
• 页码：612-630
• 全文大小：387 K

文摘

In this paper we deal with Banach spaces of analytic functions X   defined on the unit disk satisfying that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=9e41b59e0d39d55b3e17c56f5a20b3f0" title="Click to view the MathML source">R_tf∈Xclass="mathContainer hidden">class="mathCode">$R_{t}f\in X$ for any class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si2.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=653e962132a2a043da43c6f2e2e862fd" title="Click to view the MathML source">t>0class="mathContainer hidden">class="mathCode">$t>0$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si3.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=7cce2c1e426ebd5343134f03376773ba" title="Click to view the MathML source">f∈Xclass="mathContainer hidden">class="mathCode">$f\in X$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si4.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=d3b24d9a9134cf0350e164af3f95adce" title="Click to view the MathML source">R_tf(z)=f(e^itz)class="mathContainer hidden">class="mathCode">$R_{t}f(z)=f(e^{it}z)$. We study the space of functions in X   such that class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si5.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=df60d8b0f633c3b9ac9117bb542a8bb6">class="imgLazyJSB inlineImage" height="22" width="174" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304012-si5.gif">class="mathContainer hidden">class="mathCode">${\Vert P_r(Df)\Vert }_X=O(\frac{\omega (1-r)}{1-r})$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si6.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=2dfc7f7d4bd29e0478efef28a572f697" title="Click to view the MathML source">r→1⁻class="mathContainer hidden">class="mathCode">$r\to 1^{-}$ where class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si7.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=c49ccaa1c22b69a6d7af9a00baf09b9e">class="imgLazyJSB inlineImage" height="18" width="193" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304012-si7.gif">class="mathContainer hidden">class="mathCode">$Df(z)={\sum }_{n=0}^{\infty }(n+1)a_n{z}^n$ and ω is a continuous and non-decreasing weight satisfying certain mild assumptions. The space under consideration is shown to coincide with the subspace of functions in X   satisfying any of the following conditions: (a) class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si8.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=6c7de8a366cabfa20757bdd49ae6b3e7" title="Click to view the MathML source">‖R_tf−f‖_X=O(ω(t))class="mathContainer hidden">class="mathCode">${\Vert R_{t}f-f\Vert }_X=O(\omega (t))$, (b) class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si88.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=8fc6135e1c34e45209f79b0c4a94a141" title="Click to view the MathML source">‖P_rf−f‖_X=O(ω(1−r))class="mathContainer hidden">class="mathCode">${\Vert P_{r}f-f\Vert }_X=O(\omega (1-r))$, (c) class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si10.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=526fe1e29d061c1eaae46d74132ff9cd" title="Click to view the MathML source">‖Δ_nf‖_X=O(ω(2⁻ⁿ))class="mathContainer hidden">class="mathCode">${\Vert {\mathrm{\Delta }}_{n}f\Vert }_X=O(\omega (2^{-n}))$, or (d) class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si11.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=33571ec83d8f9cdc78759394ed6a567c" title="Click to view the MathML source">‖f−s_nf‖_X=O(ω(n⁻¹))class="mathContainer hidden">class="mathCode">${\Vert f-s_{n}f\Vert }_X=O(\omega (n^{-1}))$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si12.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=8e19c1accea6fd7969f55a0b1095afb8" title="Click to view the MathML source">P_rf(z)=f(rz)class="mathContainer hidden">class="mathCode">$P_{r}f(z)=f(rz)$, class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si13.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=c910845d630e4641062d2a5314de0384">class="imgLazyJSB inlineImage" height="19" width="146" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304012-si13.gif">class="mathContainer hidden">class="mathCode">$s_{n}f(z)={\sum }_{k=0}^n{a}_k{z}^k$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si14.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=8ace7f9b4b1464072a3a84b239ed393c" title="Click to view the MathML source">Δ_nf=s_2ⁿf−s_2ⁿ⁻¹fclass="mathContainer hidden">class="mathCode">${\mathrm{\Delta }}_{n}f=s_{2^n}f-s_{2^{n-1}}f$. Our results extend those known for Hardy or Bergman spaces and power weights class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304012&_mathId=si15.gif&_user=111111111&_pii=S0022247X16304012&_rdoc=1&_issn=0022247X&md5=56e302821347adb809361e9a188ad623" title="Click to view the MathML source">ω(t)=t^αclass="mathContainer hidden">class="mathCode">$\omega (t)=t^{\alpha }$.