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Estimates for the Bergman and Szegö projections in two symmetric domains of $$\mathbb{C}^ n$$. (English) Zbl 0863.47018
For $$p\geq 1$$ 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.

##### 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)
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