The Duren-Carleson theorem in tube domains over symmetric cones (Q524946)

The Duren-Carleson theorem in the setting of tube domains over symmetric cones is analyzed and some applications to pointwise multipliers are provided. Given \(T_\Omega= V+ i\Omega\) the tube domain over an irreductible symmetric cone \(\Omega\) in the complexification of a Euclidean space \(V\) of dimension \(n\) and denoting by \(r\) the rank of the cone and by \( \Delta\) the determinant function (for instance for \(V=\mathbb R^n\), \(\Omega=\Lambda_n=\{y\in \mathbb R^n: y_1^2-y_2^2-\cdots -y_n^2>0, y_1>0\}\), \(r=2\) and \(\Delta(y)=y_1^2-y_2^2-\cdots -y_n^2\)) the authors denote, for each \(0<q<\infty\) and \(\nu\in \mathbb R\), by \(L^q_\nu(T_\Omega)\) the weighted Lebesgue space with weight \(\Delta^{\eta-\frac{n}{r}}(y)dxdy\) defined on \(T_\Omega\), and by \(A^q_\nu(T_\Omega)\) the corresponding weighted Bergman space, that is, the corresponding subspace of holomorphic functions, which turns out to be non-trivial only for \(\nu>\frac{n}{r}-1\). As usual, for \(V=\mathbb R^n\), \(H^p(T_\Omega)\) stands for the space of holomorphic functions such that \(\sup_{t\in \Omega} \int_{\mathbb R^n} |f(x+it)|^pdx<\infty\). The paper deals with the description of those positive Borel measures \(\mu\) on \(T_\Omega\) such that \(H^p(T_\Omega)\) is continuously embedded into \(L^q(T_\Omega,d\mu)\). The authors give several results in this direction. First they show that, for \(0<p<q\), under certain assumptions on \(r\) and \(\nu\) (namely \(q/p>2-r/n\) and \((\nu+n/r))p/q>2n/r-1\)) the embedability of \(H^p(T_\Omega)\) into \(A^q_{n/r(q/p-1)}(T_\Omega)\) is actually equivalent to the fact that \(H^p(T_\Omega)\) embeds into \(L^q(T_\Omega,d\mu)\) for measures \(\mu\) satisfying the Carleson condition appearing when testing with the function \(\Big(\Delta^{-\nu-n/r}((z-\bar w)/2i)\Big)^{1/q}\). Then the authors analyze several cases of inclusions of Hardy spaces in Bergman spaces, extending the classical Hardy-Littlewood theorems to this setting. In particular, for \(p\geq 4\), they show the inclusion of \(H^2(T_\Omega)\) in \(A^p_{n/r(p/2-1)}(T_\Omega)\) and manage to go below \(p=4\) in the case \(\Omega=\Lambda_n\) for several values of \(n\). Finally some applications to the description of pointwise multipliers for general tubes in the particular case \(\Lambda_n\) are given.