Gradient estimates and Jackson's theorem in \(Q_\mu \) spaces related to measures
Let \(D\) be a circular starlike domain in \(\mathbb{C}^n\), i.e., 1) \(e^{i\theta} z\in D\) for all \(z\in D\) and \(0\leq\theta< 2\pi\), and 2) \(\lambda z\in D\) for all \(z\in D\), \(\lambda\in\mathbb{C}\) with \(|\lambda|< 1\). Let \(\{\mu_a\}_{a\in\Gamma}\) be a family of some restricted nonnegative \(\sigma\)-finite Borel measures in \(D\). The space \(Q_\mu= Q_\mu(D)= Q_{\mu,p}(D)\) \((0< p<\infty)\) contains the holomorphic functions in \(D\) satisfying \[ \| f\|_{Q_\mu}= \sup_{a\in\Gamma}\,\Biggl\{\int_D |\nabla f(z)|^p d\mu_a(z)\Biggr\}^{1/p}< \infty, \] where \(\nabla f(z)= \big({\partial f\over\partial z_1}(z),\dots,{\partial f\over\partial z_n}(z)\big)\) is the complex gradient of \(f\) at \(z\). The authors generalize Jackson's theorem as follows: For any \(f\in Q_p\) and \(k\in\mathbb{N}\), we have \(E_k(f,Q_\mu)\leq C(k)\omega(1/k,f,Q_\mu)\), where \(C(k)\) is a constant, \[ \omega(1/k,f,Q_\mu)= \sup_{|k-\theta'|<1/k}\| f(e^{i\theta'}z)- f(e^{i\theta''}z)\|_{Q_\mu}, \] and \(E_k(f,\theta_\mu)= \inf\| f-P_k\|\), where the infimum is taken over all polynomials of degree at most \(k\). The result covers many known spaces in the unit ball in \(\mathbb{C}^n\), and it is proved in terms of the complex gradient of \(f- P_k\).
scientific article; zbMATH DE number 5501960