The Bergman projection on vector-valued \(L^2\)-spaces with operator-valued weights
The celebrated Hunt-Muckenhoupt-Wheeden theorem characterizes the measures \(\mu\) for which the Riesz projection (or equivalently, the Hilbert transform) is bounded on \(L^2(\mathbb{R},\mu)\) (respectively, \(L^2(\mathbb{T},\mu)\)). \textit{S. Treil} and \textit{A. Volberg} [J. Funct. Anal. 143, No. 2, 269--308 (1997; Zbl 0876.42027)] used wavelet techniques to obtain an analogue of this theorem for matrix-valued weights. They show that if \(W\) is a \(d\times d\) matrix-valued weight, then the Riesz projection is bounded on \(L^2(\mathbb{R},W)\) if and only if \[ \sup_{I}\|\left(\frac{1}{|I|}\int_{I}W\right)^{1/2}\left(\frac{1}{|I|}\int_{I}W^{-1}\right)^{1/2}\|<\infty, \] where the supremum is taken over all intervals \(I\subset\mathbb{R}\). It is known that this result does not generalize to the case of operator-valued weights (that is, the case when \(d=\infty\)). The Bergman space analogue of the scalar Hunt-Muckenhoupt-Wheeden theorem was obtained by \textit{D. Békollé} and \textit{A. Bonami} [C. R. Acad. Sci., Paris, Sér. A 286, 775--778 (1978; Zbl 0398.30006)]. They characterized the measures \(\mu\) for which the Bergman projection is bounded on \(L^2(\mathbb{D},\mu)\). The paper under review, via a different approach, generalizes Békollé-Bonami's result to operator-valued weights. Before stating their result, we need some definitions. Let \(H\) be a separable Hilbert space. A weight is a function \(W:\mathbb{D}\rightarrow\mathcal{B}(H)\) such that \(W(z)\) is a nonnegative operator which is invertible for a.e. \(z\in\mathbb{D}\). For any real number \(\eta>-1\), the usual weighted Lebesgue area measure has the form \(dA_{\eta}(z)=(\eta+1)(1-|z|^2)^{\eta}dA(z)\). Given \(W\) and \(\eta\), the space \(L^2(WdA_{\eta})\) consists of \(H\)-valued functions \(f\) on \(\mathbb{D}\) which satisfy \[ \|f\|^2 = \int_{\mathbb{D}}\langle W(z)f(z),f(z)\rangle dA_{\eta}(z)<\infty. \] The Bergman projection \(P_{\eta}\) is defined on \(L^2(WdA_{\eta})\) by \[ P_{\eta}f(z)=\int_{\mathbb{D}}\frac{f(\zeta)}{(1-\bar{\zeta}z)^{\eta+2}}dA_{\eta}(z). \] The main result of the paper shows, under certain standard assumptions on \(W\), that \(P_{\eta}\) is bounded if and only if \[ \sup_{S}\|\big(\frac{1}{A_{\eta}(S)}\int_{S}WdA_{\eta}\big)^{1/2}\big(\frac{1}{A_{\eta}(S)}\int_{S}W^{-1}dA_{\eta}\big)^{1/2}\|<\infty, \] where the supremum is taken over all Carleson squares \(S\) in \(\mathbb{D}\).
scientific article; zbMATH DE number 6017647