On strict inclusions of weighted Dirichlet spaces of monogenic functions (Q2748265)

In the last years a new class of holomorphic functions, the scale of \(Q_p\)-spaces was studied, where these spaces are defined for \(0< p<\infty\) in the following way NEWLINE\[NEWLINEQ_p= \Biggl\{f:f\text{ holomorphic in the unit disk \(\Delta\) and }\sup_{a\in\Delta} \int_\Delta|f'(z)|^2 g^p(z, a) dx dy< \infty\Biggr\}.NEWLINE\]NEWLINE Here, \(g(z,a)\) is defined as \(\ln|\varphi_a(z)^{-1}|\) and \(\varphi_a= \varphi_a(z)\) is the Möbius transformation \(\varphi_a(z)= {a-z\over 1-\overline az}\). The goal of the authors is to propose hypercomplex generalizations of \(Q_p\)-spaces. First, they introduce these weighted spaces of monogenic functions in \(\mathbb{R}^3\), where the weight depends similar to the complex case on a special fundamental solution of Laplacian in \(\mathbb{R}^3\) and on its composition with a Möbius transformation. One interesting question for the obtained scale \(\{Q_p\}_{0< p< 3}\) is that for strict inclusions. Using properties of special monogenic polynomials the authors prove \(Q_p\subset Q_q\) is strict for \(0< p< q\leq 2\). For \(p>2\) all spaces \(Q_p\) coincide. The ideas used can be generalized to higher dimensions, too.