Embedding derivatives of \({\mathcal M}\)-harmonic functions into \(L^p\) spaces
A function on the unit ball \(B\subset\mathbb{C}^n\) is called \({\mathcal M}\)-harmonic if it is harmonic with respect to the invariant Laplacian on \(B\). An \({\mathcal M}\)-harmonic function \(u\) is said to belong to the \({\mathcal M}\)-harmonic Hardy space \({\mathcal H}^p\), \(0<p <\infty\), if \(M_\alpha u\in L^p (\sigma)\) for some \(\alpha>0\), where \[ M_\alpha u(\xi)= \sup\biggl\{\bigl | u(z)\bigr | :z\in D_\alpha (\xi)\biggr\} \] is the maximal function of \(u\) over the Koranyi approach region \[ D_\alpha(\xi)= \biggl\{z= p\eta\in B: \bigl| 1-\langle\eta, \xi\rangle \bigr| <\alpha (1-r)\biggr\} \] at \(\xi\in\partial B\), and \(\sigma\) is the normalized Lebesgue measure on \(\partial B\). Let \(\nabla^\mu\) be the gradient of order \(m\geq 1\) for the function \(u\), \[ | \nabla^mu |^2= \sum_{| \alpha|+ |\beta |=m} \left(\left | {\partial^\alpha u\over\partial z^\alpha_u} \right|^2 +\left| {\partial^\beta u\over \partial\overline z^\beta} \right|^2 \right). \] The author obtains a characterization for those Borel measure \(\mu\) on \(B\) such that \(\nabla^mu\in L^q(\mu)\) whenever \(u\in{\mathcal H}^p\), assuming \(0<q<p<\infty\). The case \(0<p\leq q<\infty\) was studied earlier by M. Jevtić.
scientific article; zbMATH DE number 1309609