Compact embedding derivatives of Hardy spaces into Lebesgue spaces (Q2790186)

For any \(0<p,q<\infty\) and \(n\in\mathbb{N}\), the author characterizes the positive Borel measures \(\mu\) on the open unit disk \(\mathbb{D}\) in \(\mathbb{C}\) such that the inclusion mapping from the Hardy-Sobolev space \(H^p_{-n}(\mathbb{D})=\{f^{(n)}:\,f\in H^p(\mathbb{D})\}\) into \(L^q(\mathbb{D},d\mu)\) is compact. This characterization is given in terms of the function NEWLINE\[NEWLINE\Phi_\mu(z):=(1-|z|^2)^{-1-nq}\mu(\Delta(z,r)),NEWLINE\]NEWLINE where \(\Delta(z,r)\) denotes the pseudohyperbolic disk of center \(z\) and radius \(r\).NEWLINENEWLINEThe main result in this paper proves that if \(D^{(n)}\) is the differentiation operator of order \(n\geq 1\), then \(D^{(n)}:H^p(\mathbb{D})\to L^q(\mathbb{D},d\mu)\) is compact if and only if:NEWLINENEWLINE1) If \(q<\min\{2,p\}\), for any fixed \(r\in(0,1)\) the function \(\Phi_\mu\) is in the tent space NEWLINE\[NEWLINET^{p/(p-q)}_{2/(2-q)}\bigl(dA(z)/(1-|z|^2)^2\bigr).NEWLINE\]NEWLINENEWLINENEWLINE2) If \(q=p<2\), NEWLINE\[NEWLINE \lim_{|w|\to 1^{-}}\frac {1}{1-|w|^2}\int_{T_w} \Phi_\mu(z)^{2/(2-p)}\frac{dA(z)}{1-|z|^2}=0, NEWLINE\]NEWLINE where \(T_w\) is a tent associated to the point \(w\in\mathbb{D}\).NEWLINENEWLINE3) If \(2\leq q<p\), NEWLINE\[NEWLINE \lim_{R\to 1^{-}}\int_{|\zeta|=1}\left(\sup_{z\in \Gamma_\zeta\setminus \overline{D(0,R)}} \Phi_\mu(z)\right)^{p/(p-q)} d\sigma (\zeta)=0, NEWLINE\]NEWLINE where \(\Gamma_\zeta\) denotes a Stolz angle of the vertex \(\zeta\), and \(D(0,R)\) is the Euclidean disk of center \(0\) and radius \(R\).NEWLINENEWLINE4) If either \(q>p\) or \(2\leq q=p\), then \((1-|z|^2)^{1-q/p}\Phi_\mu(z)\to 0\) as \(|z|\to 1^{-}\). This is also equivalent to \((1-|z|^2)^{-q/p-nq}\mu(S_z)\to 0\) as \(|z|\to 1^{-}\), where \(S_z\) is a Carleson square associated to \(z\in\mathbb{D}\).