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Carleson embeddings. (English) Zbl 0937.47024
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 R. R. Coifman, Y. Meyer and 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.
##### MSC:
 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.) 47B33 Linear composition operators 47B38 Linear operators on function spaces (general) 46E15 Banach spaces of continuous, differentiable or analytic functions
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