Boundedness of Bergman projections on tube domains over light cones. (English) Zbl 0983.32001
Let \(\Gamma=\{y\in \mathbb{R}^n: y_n>(y^2_1+ \cdots+y^2_{n-1})^{1/ 2}\}\) be the future light cone, \(\Omega=\mathbb{R}^n+i\Gamma\) be the associated tube domain, and \(Q(z)= -z^2_1- \cdots-z^2_{n-1} +z_n^2\) be the Lorentz quadratic form. The weighted Bergman space \(A^p_0\) is the closed subspace of \(L^p_\nu =L^p (\Omega,Q(y)^{\nu-n} dxdy)\) consisting of holomorphic functions. The weighted Bergman kernel \(B_\nu(z,w)= c_\nu Q(z- \overline w)^{-\nu}\) is the reproducing kernel on \(A^2_\nu\), and the weighted Bergman projection
\[
P_\nu f(z)=\int_\Omega f(u+iv) B_\nu(z,u+iv) Q(\nu)^{\nu-n}dudv
\]
is the orthogonal projection of \(L^2_\nu\) onto \(A^2_\nu\).
Using technique of mixed norm spaces and Laplace-Fourier transforms, the authors prove that the operator \(P_\nu\) is bounded on \(L^p_\nu\) if \[ 1+{n-2\over 2(\nu-1)}< p<1+{2(\nu-1)\over n-2}. \]
Using technique of mixed norm spaces and Laplace-Fourier transforms, the authors prove that the operator \(P_\nu\) is bounded on \(L^p_\nu\) if \[ 1+{n-2\over 2(\nu-1)}< p<1+{2(\nu-1)\over n-2}. \]
Reviewer: Alexandr Yu.Rashkovsky (Khar’kov)
MSC:
32A07 | Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010) |
32A36 | Bergman spaces of functions in several complex variables |
32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |
42B35 | Function spaces arising in harmonic analysis |