Theorems on the embedding of Bergman spaces in Lebesgue spaces for domains with nonsmooth boundary (Q679505)

For an open set \(G\subset\mathbb{C}\), \(G\neq\mathbb{C}\), and a positive Borel measure \(\mu\) on \(G\), let \(d(z)=\text{dist}(z,\partial G)\), \(z\in G\), \(\mu(z)= \mu(B(z,d(z)/2))\), where \(B(z,r)\) is the open disk in \(\mathbb{C}\) with center \(z\in\mathbb{C}\) and radius \(r>0\). Let \(dm_\beta= d(z)^\beta dm\), where \(m\) is the plane Lebesgue measure and \(\beta>-1\). The Bergman space \(L^p_a(G,\mu)\) is defined to be the subspace of \(L^p(G,\mu)\) consisting of analytic functions. The authors give sufficiently conditions for a domain \(G\) that for any \(p>0\), \(q>0\) and a nonnegative Borel measure \(\mu\) on \(G\) the following conditions are equivalent: (i) \(\mu(z)m_\beta(z)^{-1}\in L^{p/(p-q)}(G,m_\beta)\) if \(q<p\), and \(\sup_{z\in G}\mu(z)m_\beta(z)^{-q/p}<\infty\) if \(q\geq p\); (ii) the embedding operator \(I:L^p_a(G,m_\beta)\subset L^q(G,\mu)\) is bounded.