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Hp spaces and integral operators

Aleman, Alexandru LU (1987) In Mathematica (Cluj) 29(52)(2). p.99-104
Abstract
Let $\scr H(U)$ denote the class of analytic functions in the unit disc $U$ and $g$ be analytic in $U$, normalized by $g(0)=g'(0)-1=0$ and $g(z)\ne0$ for $z\in U\sbs\{0\}$. $H^p$, $0<p\le \infty$, denotes the Hardy class and $H\,\roman{log}^+\,H$ the class for which $\int_0^{2\pi}|f(re^{i\theta})| \roman{log}^+|f(re^{i\theta})|\,d\theta$ is bounded when $r\rightarrow 1^-$. The author considers the integral operator $L_g\colon \scr H(U)\rightarrow \scr H(U)$ defined by $L_g(f)(z)=(z/g(z))\int_0^{z}f(t)g'(t)\,dt$ and shows that: (i) if $zg'/g\in H\,\roman{log}^+\,H$ and $f\in H^p$ then $L_g(f)\in H^p$; (ii) if $zg'/g\in H^q$, $q>1$ and $f\in H^p$ then $L_g(f)\in H^r$ where $r=pq/(p+q-pq)$ for $0<p<q/(q-1)$ and $r=\infty$ for... (More)
Let $\scr H(U)$ denote the class of analytic functions in the unit disc $U$ and $g$ be analytic in $U$, normalized by $g(0)=g'(0)-1=0$ and $g(z)\ne0$ for $z\in U\sbs\{0\}$. $H^p$, $0<p\le \infty$, denotes the Hardy class and $H\,\roman{log}^+\,H$ the class for which $\int_0^{2\pi}|f(re^{i\theta})| \roman{log}^+|f(re^{i\theta})|\,d\theta$ is bounded when $r\rightarrow 1^-$. The author considers the integral operator $L_g\colon \scr H(U)\rightarrow \scr H(U)$ defined by $L_g(f)(z)=(z/g(z))\int_0^{z}f(t)g'(t)\,dt$ and shows that: (i) if $zg'/g\in H\,\roman{log}^+\,H$ and $f\in H^p$ then $L_g(f)\in H^p$; (ii) if $zg'/g\in H^q$, $q>1$ and $f\in H^p$ then $L_g(f)\in H^r$ where $r=pq/(p+q-pq)$ for $0<p<q/(q-1)$ and $r=\infty$ for $p\ge\break q/(q-1)$; and (iii) if $zg'/g\in H^\infty$ and $f\in H^p$, then $L_g(f)$ is in $H^r$ where $r=p/(1-p)$ for $0<p<1$ and $r=\infty$ for $p\ge 1$. This result generalizes a result of the reviewer [same journal 29(52) (1987), no. 1, 29--31; MR0939548 (89e:30061)]. An interesting example is given. (Less)
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author
publishing date
type
Contribution to journal
publication status
published
subject
in
Mathematica (Cluj)
volume
29(52)
issue
2
pages
99 - 104
publisher
Editions de l'Academie Roumaine
ISSN
1222-9016
language
English
LU publication?
no
id
fa48100a-8c98-4515-82a7-b8d88acb3c3f (old id 1467384)
date added to LUP
2009-09-16 14:30:53
date last changed
2016-06-29 09:06:48
@misc{fa48100a-8c98-4515-82a7-b8d88acb3c3f,
  abstract     = {Let $\scr H(U)$ denote the class of analytic functions in the unit disc $U$ and $g$ be analytic in $U$, normalized by $g(0)=g'(0)-1=0$ and $g(z)\ne0$ for $z\in U\sbs\{0\}$. $H^p$, $0&lt;p\le \infty$, denotes the Hardy class and $H\,\roman{log}^+\,H$ the class for which $\int_0^{2\pi}|f(re^{i\theta})| \roman{log}^+|f(re^{i\theta})|\,d\theta$ is bounded when $r\rightarrow 1^-$. The author considers the integral operator $L_g\colon \scr H(U)\rightarrow \scr H(U)$ defined by $L_g(f)(z)=(z/g(z))\int_0^{z}f(t)g'(t)\,dt$ and shows that: (i) if $zg'/g\in H\,\roman{log}^+\,H$ and $f\in H^p$ then $L_g(f)\in H^p$; (ii) if $zg'/g\in H^q$, $q&gt;1$ and $f\in H^p$ then $L_g(f)\in H^r$ where $r=pq/(p+q-pq)$ for $0&lt;p&lt;q/(q-1)$ and $r=\infty$ for $p\ge\break q/(q-1)$; and (iii) if $zg'/g\in H^\infty$ and $f\in H^p$, then $L_g(f)$ is in $H^r$ where $r=p/(1-p)$ for $0&lt;p&lt;1$ and $r=\infty$ for $p\ge 1$. This result generalizes a result of the reviewer [same journal 29(52) (1987), no. 1, 29--31; MR0939548 (89e:30061)]. An interesting example is given.},
  author       = {Aleman, Alexandru},
  issn         = {1222-9016},
  language     = {eng},
  number       = {2},
  pages        = {99--104},
  publisher    = {ARRAY(0x9f435a8)},
  series       = {Mathematica (Cluj)},
  title        = {Hp spaces and integral operators},
  volume       = {29(52)},
  year         = {1987},
}