It is shown in quantitative terms that the maximal Bergman projection

$$ {P}^{+}_{\omega}(f)(z)={\int}_{\mathbb{D}} f(\zeta)|{B}^{\omega}_{z}(\zeta)|\omega(\zeta) dA(\zeta), $$

is bounded from \(L^{p}_{\nu }\) to \(L^{p}_{\eta }\) if and only if

$$ \underset{0<r<1}{\sup}\left( {\int}_{0}^{r}\frac{\eta(s)}{\left( {\int}_{s}^{1}\omega(t) dt\right)^{p}} ds+1\right)^{\frac{1}{p}} \left( {\int}_{r}^{1}\left( \frac{\omega(s)}{\nu(s)^{\frac{1}{p}}}\right)^{p^{\prime}}ds\right)^{\frac{1}{p^{\prime}}}<\infty, $$

provided ω,ν,η are radial regular weights. A radial weight σ is regular if it satisfies \({\sigma }(r){\asymp {\int \limits }}_{r}^{1}{\sigma }(t) dt/(1-r)\) for all 0 ≤ r < 1. It is also shown that under an appropriate additional hypothesis involving ω and η, the Bergman projection P_ω and \({P}^{+}_{\omega }\) are simultaneously bounded.

中文翻译：

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由规则权重引起的最大Bergman投影的径向两个权重不等式

定量表示最大伯格曼投影

$$ {P} ^ {+} _ {\ omega}（f）（z）= {\ int} _ {\ mathbb {D}} f（\ zeta）| {B} ^ {\ omega} _ {z }（\ zeta）| \ omega（\ zeta）dA（\ zeta），$$

仅当且仅当从\（L ^ {p} _ {\ nu} \）到\（L ^ {p} _ {\ eta} \）范围内

$$ \ underset {0 <r <1} {\ sup} \ left（{\ int} _ {0} ^ {r} \ frac {\ eta（s）} {\ left（{\ int} _ {s } ^ {1} \ omega（t）dt \ right）^ {p}} ds + 1 \ right）^ {\ frac {1} {p}} \ left（{\ int} _ {r} ^ {1 } \ left（\ frac {\ omega（s）} {\ nu（s）^ {\ frac {1} {p}}} \ right）^ {p ^ {\ prime}} ds \ right）^ {\ frac {1} {p ^ {\ prime}}} <\ infty，$$

假设ω，ν，η是径向规则权重。如果径向权重σ满足\（{\ sigma}（r）{\ asymp {\ int \ limits}} _ {r} ^ {1} {\ sigma}（t）dt /（1-r） \）对于所有0≤ [R <1。还表明，在涉及一个适当的附加假设ω和η，所述伯格曼投影P _ω和\（{P} ^ {+} _ {\欧米加} \）被同时限定。