Carleson embeddings (Q1809719)

This is a paper on special spaces of functions \(\mathbb{D}\to\mathbb{C}\) where \(\mathbb{D}\) is the open unit disk in \(\mathbb{C}\). Let \(\mathbb{T}\) be the boundary of \(\mathbb{D}\), and let \(m\) be normalized Lebesgue measure on \(\mathbb{T}\). Put \(B(0)= \mathbb{T}\), and if \(0\neq z\in\mathbb{D}\), let \(B(z)\) be the arc of length \(1-|z|^2\) centered at \(z/|z|\). The Stoltz domain for \(\zeta\in\mathbb{T}\) is \(\Gamma(\zeta)= \{z\in\mathbb{D}: \zeta\in B(z)\}\), and the tent over an open set \(\Omega\subset\mathbb{T}\) is \(\theta(\Omega)= \Omega\cup \{z\in\mathbb{D}: B(z)\subset \Omega\}\). Following \textit{R. R. Coifman}, \textit{Y. Meyer} and \textit{E. M. Stein} [J. Funct. Anal. 62, 304-335 (1985; Zbl 0569.42016)] the non-tangential supremum \(Nf(\zeta):= \sup_{z\in\Gamma(\zeta)}|f(z)|\) of a function \(f:D\to \mathbb{C}\) at \(\zeta\in\mathbb{T}\) is used to define, for \(0<q\leq\infty\), the `tent space' \(T^q(D)\) as the closure of \({\mathcal C}(\overline D)\) in the (\(q\)-)Banach space of all \(f:\mathbb{D}\to \mathbb{C}\) such that \(\|Nf\|_{L^q(m)}\) is finite. For \(1<q<\infty\) (resp. \(0<q<\infty\)), the usual harmonic space \(h^q(\mathbb{D})\) (Hardy space \(H^q(\mathbb{D}))\) just consists of all harmonic (analytic) members of \(T^q(\mathbb{D})\). Given \(\beta>0\), say that a (regular) Borel measure \(\mu\) on \(\overline{\mathbb{D}}\) is a \(\beta\)-Carleson measure if there is a constant \(C\) such that \(|\mu|(\theta(\Omega))\leq C\cdot m(\Omega)^\beta\) for every open set \(\Omega\subset\mathbb{T}\). Let \(\|\mu\|_\beta\) be the smallest of such constants. The \(\beta\)-Carleson measures form a Banach space \(M^\beta\) with norm \(\|\cdot\|_\beta\); it is isometrically isomorphic to the dual of the tent space \(T^{1/\beta}(\mathbb{D})\). Moreover, the formal identity \(I_\mu: T^q(\mathbb{D})\hookrightarrow L^{\beta q}(\mu)\) exists and is continuous if and only if \(\mu\) is in \(M^\beta\). In the light of the fact that the image measure \(m\circ \phi^{-1}\) of an analytic self-map \(\Phi:\mathbb{D}\to \mathbb{D}\) is a \(\beta\)-Carleson measure if and only if the composition operator \(C_\phi: f\mapsto f\circ\phi\) maps \(H^q\) into \(H^{\beta q}\) (independent of \(q\)), this leads to generalizations of several known results on composition operators. Particular emphasis is put on the question under which conditions \(I_\mu\) is compact, weakly compact, \(p\)-integral, absolutely \(p\)-summing, etc.