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 R_tf∈X$R_{t}f\in X$ for any t>0$t>0$ and f∈X$f\in X$, where R_tf(z)=f(e^itz)$R_{t}f(z)=f(e^{it}z)$. We study the space of functions in X   such that 9117bb542a8bb6">${\Vert P_r(Df)\Vert }_X=O(\frac{\omega (1-r)}{1-r})$, r→1⁻$r\to 1^{-}$ where $Df(z)={\sum }_{n=0}^{\infty }(n+1)a_n{z}^n$ 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) ‖R_tf−f‖_X=O(ω(t))${\Vert R_{t}f-f\Vert }_X=O(\omega (t))$, (b) ‖P_rf−f‖_X=O(ω(1−r))${\Vert P_{r}f-f\Vert }_X=O(\omega (1-r))$, (c) ‖Δ_nf‖_X=O(ω(2⁻ⁿ))${\Vert {\mathrm{\Delta }}_{n}f\Vert }_X=O(\omega (2^{-n}))$, or (d) ‖f−s_nf‖_X=O(ω(n⁻¹))${\Vert f-s_{n}f\Vert }_X=O(\omega (n^{-1}))$, where P_rf(z)=f(rz)$P_{r}f(z)=f(rz)$, $s_{n}f(z)={\sum }_{k=0}^n{a}_k{z}^k$ and Δ_nf=s_2ⁿf−s_2ⁿ⁻¹f${\mathrm{\Delta }}_{n}f=s_{2^n}f-s_{2^{n-1}}f$. Our results extend those known for Hardy or Bergman spaces and power weights ω(t)=t^α$\omega (t)=t^{\alpha }$.