由加倍权诱导的 Lebesgue 空间上的 Bergman 投影
复变函数
2023-06-16 v1 泛函分析
摘要
设 ω \omega ω ν \nu ν σ = ω p ′ ν − p ′ p \sigma=\omega^{p'}\nu^{-\frac{p'}p} σ = ω p ′ ν − p p ′ ω x = ∫ 0 1 s x ω ( s ) d s \omega_x =\int_0^1 s^x \omega(s)\,ds ω x = ∫ 0 1 s x ω ( s ) d s 1 ≤ x < ∞ 1\le x<\infty 1 ≤ x < ∞ ω \omega ω P ω P_\omega P ω D p ( ω , ν ) = sup n ∈ N ∪ { 0 } ( ν n p + 1 ) 1 p ( σ n p ′ + 1 ) 1 p ′ ω 2 n + 1 < ∞ D_p(\omega,\nu)=\sup_{n\in \mathbb{N}\cup\{0\}}\frac{\left(\nu_{np+1}\right)^\frac1p\left(\sigma_{np'+1}\right)^\frac1{p'}}{\omega_{2n+1}}<\infty D p ( ω , ν ) = n ∈ N ∪ { 0 } sup ω 2 n + 1 ( ν n p + 1 ) p 1 ( σ n p ′ + 1 ) p ′ 1 < ∞ ν \nu ν sup 0 ≤ r < 1 ∫ r 1 ω ( s ) s d s ∫ 1 + r 2 1 ω ( s ) s d s < ∞ \sup_{0\le r<1}\frac{\int_r^1 \omega(s)s\,ds}{\int_{\frac{1+r}{2}}^1 \omega(s)s\,ds}<\infty sup 0 ≤ r < 1 ∫ 2 1 + r 1 ω ( s ) s d s ∫ r 1 ω ( s ) s d s < ∞ sup 0 ≤ r < 1 ∫ r 1 ω ( s ) s d s ∫ r 1 − 1 − r K ω ( s ) s d s < ∞ \sup_{0\le r<1}\frac{\int_r^1 \omega(s)s\,ds}{\int_r^{1-\frac{1-r}{K}} \omega(s)s\,ds}<\infty sup 0 ≤ r < 1 ∫ r 1 − K 1 − r ω ( s ) s d s ∫ r 1 ω ( s ) s d s < ∞ K > 1 K>1 K > 1 D p ( ω , ν ) < ∞ D_p(\omega,\nu)<\infty D p ( ω , ν ) < ∞ ∥ P ω ( f ) ∥ D ν , k p ≤ C ∥ f ∥ L ν p , \|P_\omega(f)\|_{D^p_{\nu,k}}\le C\|f\|_{L^p_\nu}, ∥ P ω ( f ) ∥ D ν , k p ≤ C ∥ f ∥ L ν p , ∥ f ∥ D ν , k p p = ∑ j = 0 k − 1 ∣ f ( j ) ( 0 ) ∣ p + ∫ D ∣ f ( k ) ( z ) ∣ p ( 1 − ∣ z ∣ ) k p ν ( z ) d A ( z ) < ∞ , k ∈ N , \Vert f \Vert_{D^p_{\nu, k}}^p =\sum\limits_{j=0}^{k-1}| f^{(j)}(0)|^p+ \int_{\mathbb{D}} \vert f^{(k)}(z)\vert^p (1-|z| )^{kp}\nu(z)\,dA(z)<\infty, \quad k\in \mathbb{N}, ∥ f ∥ D ν , k p p = j = 0 ∑ k − 1 ∣ f ( j ) ( 0 ) ∣ p + ∫ D ∣ f ( k ) ( z ) ∣ p ( 1 − ∣ z ∣ ) k p ν ( z ) d A ( z ) < ∞ , k ∈ N , ν ( r ) = exp ( − α ( 1 − r l ) β ) \nu(r)=\exp \left(-\frac{\alpha}{(1-r^l)^{\beta}} \right) ν ( r ) = exp ( − ( 1 − r l ) β α ) ω ( r ) = exp ( − α ~ ( 1 − r l ~ ) β ~ ) \omega(r)= \exp \left(-\frac{\widetilde{\alpha}}{(1-r^{\widetilde{l}})^{\widetilde{\beta}}} \right) ω ( r ) = exp ( − ( 1 − r l ) β α ) 0 < α , α ~ , l , l ~ < ∞ 0<\alpha, \, \widetilde{\alpha}, \, l, \, \widetilde{l}<\infty 0 < α , α , l , l < ∞ 0 < β , β ~ ≤ 1 0<\beta , \, \widetilde{\beta}\le 1 0 < β , β ≤ 1 D p ( ω , ν ) < ∞ D_p(\omega,\nu)<\infty D p ( ω , ν ) < ∞
引用
@article{arxiv.2306.08255,
title = {Bergman projection on Lebesgue space induced by doubling weight},
author = {José Ángel Peláez and Elena de la Rosa and Jouni Rättyä},
journal= {arXiv preprint arXiv:2306.08255},
year = {2023}
}