作用于Bergman与Lebesgue空间的径向平均算子
复变函数
2019-09-23 v1
摘要
本文证明了由单位圆盘D \mathbb{D} D ω \omega ω T ω ( f ) ( z ) = ∫ ∣ z ∣ 1 f ( s z ∣ z ∣ ) ω ( s ) d s ω ^ ( z ) , ω ^ ( z ) = ∫ ∣ z ∣ 1 ω ( s ) d s , T_\omega(f)(z)=\frac{\int_{|z|}^1f\left(s\frac{z}{|z|}\right)\omega(s)\,ds}{\widehat{\omega}(z)},\quad \widehat{\omega}(z)=\int_{|z|}^1\omega(s)\,ds, T ω ( f ) ( z ) = ω ( z ) ∫ ∣ z ∣ 1 f ( s ∣ z ∣ z ) ω ( s ) d s , ω ( z ) = ∫ ∣ z ∣ 1 ω ( s ) d s , A ν p A^p_\nu A ν p 0 < p < ∞ 0<p<\infty 0 < p < ∞ ν \nu ν 0 ≤ r < 1 0\leq r<1 0 ≤ r < 1 ν ^ ( r ) ≤ C ν ^ ( 1 + r 2 ) \widehat{\nu}(r)\leq C\widehat{\nu}\left(\frac{1+r}{2}\right) ν ( r ) ≤ C ν ( 2 1 + r ) L ν p L^p_\nu L ν p sup 0 ≤ r < 1 ω ^ ( r ) p ∫ r 1 s ν ( s ) d s ∫ 0 r t ν ( t ) ω ^ ( t ) p d t < ∞ \sup_{0\leq r<1}\frac{\widehat{\omega}(r)^p}{\int_{r}^1 s\nu(s)\,ds}\int_0^r\frac{t\nu(t)}{\widehat{\omega}(t)^p}\,dt<\infty sup 0 ≤ r < 1 ∫ r 1 s ν ( s ) d s ω ( r ) p ∫ 0 r ω ( t ) p t ν ( t ) d t < ∞ ω \omega ω ν \nu ν η \eta η η ( { z ∈ D : ∣ T ω ( f ) ( z ) ∣ ≥ λ } ) ≲ λ − p ∥ f ∥ L ν p p , λ > 0 , \eta \left(\left\{ z\in\mathbb{D} : |T_\omega(f)(z)|\geq\lambda\right\}\right)\lesssim\lambda^{-p} \| f\|_{L^p_\nu}^p,\quad \lambda>0, η ( { z ∈ D : ∣ T ω ( f ) ( z ) ∣ ≥ λ } ) ≲ λ − p ∥ f ∥ L ν p p , λ > 0 , A ν p → L ν p A^p_\nu\to L^p_\nu A ν p → L ν p L ν p → L ν p L^p_\nu\to L^p_\nu L ν p → L ν p L ν p → L ν p , ∞ L^p_\nu\to L^{p,\infty}_\nu L ν p → L ν p , ∞
引用
@article{arxiv.1811.12745,
title = {Radial averaging operator acting on Bergman and Lebesgue spaces},
author = {Taneli Korhonen and Jose Angel Pelaez and Jouni Rattya},
journal= {arXiv preprint arXiv:1811.12745},
year = {2019}
}