Reproducing kernels and radial differential operators for holomorphic and harmonic Besov spaces on unit balls: a unified view
The author provides a short survey of the basic characteristics of the two--parameter Besov spaces \(B^p_q,\, 1\leq q <\infty, q\in\mathbb R\), (respectively \(b^p_q\)) of holomorphic (respectively harmonic) functions on the unit ball \(\mathbb B\) in \(\mathbb C^n\) (respectively \(\mathbb R^n\)). For \(q>-1\), these become the weighted Bergman spaces of holomorphic (respectively harmonic) functions in \(L^p_q(\mathbb B) = L^p(d\nu_q)\), where \(d\nu_q\) is the measure \((1-|z|^2)^qd\nu\) (respectively \((1-|x|^2)^qd\nu) \) on \(\mathbb B\). For \(q\leq -1\), the spaces \(B^p_q\) as well as \(b^p_q\) are defined in terms of the radial differential operators \(D^t_s\) of fractional order \(t\in\mathbb R\). For both holomorphic functions \(f\) and harmonic functions \(u\), the operator \(D^t_s\) is defined in terms of the expansion of \(f\) in terms of homogeneous holomorphic polynomials \(f_k(z)\), and the expansion of \(u\) in terms of homogeneous harmonic polynomials \(u_m(x)\). Thus for \(q\in\mathbb R\), \(1\leq p<\infty\), \(B^p_q\) is the space of holomorphic functions \(f\) on \(\mathbb B\) for which \((1-|z|^2)^tD^t_sf(z) \in L^p_q(\mathbb B)\), where \(t\) is chosen such that \(q+pt >-1\), with a similar definition for \(b^p_q\). The paper stresses similarities as well as contrasts between the two spaces. It also investigates some relationships between the reproducing kernels of the Hilbert spaces \(B^2_q\) and \(b^2_q\).
scientific article; zbMATH DE number 5840921
scientific article