Estimates for the Bergman and Szegö projections in two symmetric domains of \(\mathbb{C}^ n\).

*(English)*Zbl 0863.47018For \(p\geq 1\) the Bergman space of a domain (endowed with the Lebesgue measure) \(D\) in \(\mathbb{C}^n\) is \(A^p(D)=H(D)\cap L^p(D)\). The first part of this paper studies for which \(p\), \(q\) the Bergman orthogonal projector \(P:L^2\to A^2\) (associated to the Bergman kernel \(B(\cdot,\cdot)\)), or the integral operator \(P^*\) on \(L^2\) associated to the kernel \(|B(\cdot,\cdot)|\), are bounded or unbounded on \(L^p\) to \(L^q\), when \(n\geq 3\) and \(D\) is either the tube \(\Omega=\mathbb{R}^n+i\Gamma\) over the spherical cone \(\Gamma=\{y\in\mathbb{R}^n: y_1>0, y_1y_2>\sum^n_{j=3}y^2_j\}\) or the Lie ball \(\omega=\{z\in\mathbb{C}^n:2|z|^2-1<|\sum^n_{j=1}z^2_j|<1\}\). The authors prove that: on \(\Omega\) or \(\omega\) the projection \(P\) is unbounded on \(L^p\) (to itself) if \(1<p\leq 3/2-1/n\) or \(3+4/(n-2)\leq p<\infty\), while \(P^*\) is bounded on \(L^p\) if and only if \(2-2/n<p<2+2/(n-2)\), in which case \(P\) is bounded on \(L^p\) to \(A^p\); on \(\omega\) (bounded, unlike \(\Omega\)) the operator \(P^*\) on \(L^\infty\) to \(L^q\) is bounded if and only if \(q<2+4/(n-2)\), in which case the Bloch space \(\mathcal B\) of holomorphic functions on \(\omega\), being the range of \(P\) on \(L^\infty\), is contained in \(A^q\) with continuous inclusion, whereas if \(q\geq 4+8/(n-2)\) the projection \(P\) on \(L^\infty\) to \(L^q\) is bounded, and there is no continuous inclusion of \(\mathcal B\) into \(A^q\). The results on \(\omega\) are deduced from those on the holomorphically equivalent domain \(\Omega\) via a suitable transfer principle.

The second part of the paper studies for which \(p\) the Szegö orthogonal projector \(\mathbb{S}\) of \(L^2(\partial_0D)\) onto the subspace of boundary values of functions in the Hardy space \(H^2(D)\) is unbounded on \(L^p(\partial_0D)\) to \(L^q(\partial_0D)\), when \(D\) is any standard bounded realization of rank greater than 1 of the tube \(\mathbb{R}^n+i{\mathcal C}\) over a self-dual cone \(\mathcal C\) in \(\mathbb{R}^n\); the measure on the Shilov boundary \(\partial_0D\) of \(D\) is one which is invariant under the stability group of the origin. The authors prove that, on any such \(D\), the projection \(\mathbb{S}\) is unbounded on \(L^p\) if \(1<p<2\) or \(2<p<\infty\), and provide a new proof, again based on the transfer principle, that, on \(\omega\), the projection \(\mathbb{S}\) is unbounded on \(L^\infty\) to \(L^q\) if \(q\geq 2+4/(n-2)\), a result due to B. Jöricke.

The second part of the paper studies for which \(p\) the Szegö orthogonal projector \(\mathbb{S}\) of \(L^2(\partial_0D)\) onto the subspace of boundary values of functions in the Hardy space \(H^2(D)\) is unbounded on \(L^p(\partial_0D)\) to \(L^q(\partial_0D)\), when \(D\) is any standard bounded realization of rank greater than 1 of the tube \(\mathbb{R}^n+i{\mathcal C}\) over a self-dual cone \(\mathcal C\) in \(\mathbb{R}^n\); the measure on the Shilov boundary \(\partial_0D\) of \(D\) is one which is invariant under the stability group of the origin. The authors prove that, on any such \(D\), the projection \(\mathbb{S}\) is unbounded on \(L^p\) if \(1<p<2\) or \(2<p<\infty\), and provide a new proof, again based on the transfer principle, that, on \(\omega\), the projection \(\mathbb{S}\) is unbounded on \(L^\infty\) to \(L^q\) if \(q\geq 2+4/(n-2)\), a result due to B. Jöricke.

Reviewer: E.Casadio Tarabusi (Roma)

##### MSC:

47B38 | Linear operators on function spaces (general) |

32A25 | Integral representations; canonical kernels (Szegő, Bergman, etc.) |

42B99 | Harmonic analysis in several variables |

32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |

46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |