A radial weight \(\omega \) belongs to the class \(\widehat{\mathcal {D}}\) if there exists \(C=C(\omega )\ge 1\) such that \(\int _r^1 \omega (s)\,ds\le C\int _{\frac{1+r}{2}}^1\omega (s)\,ds\) for all \(0\le r<1\). Write \(\omega \in \check{\mathcal {D}}\) if there exist constants \(K=K(\omega )>1\) and \(C=C(\omega )>1\) such that \({\widehat{\omega }}(r)\ge C{\widehat{\omega }}\left( 1-\frac{1-r}{K}\right) \) for all \(0\le r<1\). These classes of radial weights arise naturally in the operator theory of Bergman spaces induced by radial weights (Peláez and Rättyä in Bergman projection induced by radial weight, 2019. arXiv:1902.09837). Classical results by Hardy and Littlewood (J Reine Angew Math 167:405–423, 1932) and Shields and Williams (Mich Math J 29(1):3–25, 1982) show that the weighted Bergman space of harmonic functions is not closed by harmonic conjugation if \({\omega \in \widehat{\mathcal {D}}\setminus \check{\mathcal {D}}}\) and \(0<p\le 1\). In this paper we establish sharp estimates for the norm of the analytic Bergman space \(A^p_\omega \), with \({\omega \in \widehat{\mathcal {D}}\setminus \check{\mathcal {D}}}\) and \(0<p<\infty \), in terms of quantities depending on the real part of the function. It is also shown that these quantities result equivalent norms for certain classes of radial weights.

中文翻译：

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倍增权在Bergman空间上的调和共轭

如果存在\（C = C（\ omega）\ ge 1 \），则径向权重\（\ omega \）属于类\（\ widehat {\ mathcal {D}} \\），使得\（\ int _r ^ 1 \ omega（s）\，ds \ le C \ int _ {\ frac {1 + r} {2}} ^ 1 \ omega（s）\，ds \）所有\（0 \ le r <1 \）。如果存在常量（（K = K（\ omega）> 1 \）和\（C = C（\ omega）> 1 \），请写\（\ omega \ in \ check {\ mathcal {D}} \）即\（{\ widehat {\欧米加}}（R）\ GE 13 C {\ widehat {\欧米加}} \左（1- \压裂{1-R} {K} \右）\）对所有\（0 \ le r <1 \）。这些类别的径向权重自然产生于由径向权重引起的Bergman空间的算符理论中（Peláez和Rättyä在由径向权重引起的Bergman投影中，2019。arXiv：1902.09837）。Hardy和Littlewood（J Reine Angew Math 167：405–423，1932）和Shields and Williams（Mich Math J 29（1）：3–25，1982）的经典结果表明，调和函数的加权Bergman空间没有关闭如果\（{\ omega \ in \ widehat {\ mathcal {D}} \ setminus \ check {\ mathcal {D}}} \）和\（0 <p \ le 1 \）则通过谐波共轭。在本文中，我们对解析Bergman空间\（A ^ p_ \ omega \）的范数建立了清晰的估计，其中\（{\ omega \ in \ widehat {\ mathcal {D}} \ setminus \ check {\ mathcal { D}}} \）和\（0 <p <\ infty \），其数量取决于函数的实部。还表明，这些数量得出了某些类别的径向重量的等效标准。