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Analogs of Carleson’s imbedding theorem for some spaces of analytic functions. (Russian. English summary) Zbl 0591.30032

The author extends the Carleson measure theorem in various ways. Sample results: Theorem 1. Let \(\mu\) be a positive Borel measure on the upper halfplane \(C_+\) and \(\phi\) a positive, concave, increasing function on \(R_+=(0,+\infty)\). Then the transform \[ ({\mathcal F}f)(z)=\int^{\infty}_{0}f(t)e^{itz}dt \] defined in the space \(L^ 2(R_+,\phi (t)^{-1}dt)\) with values in the space \(L^ 2(C_ 0,\mu)\) has the norm comparable to \(\sup \{\epsilon^{-1}\phi (\epsilon^{-1})\mu (K_{x,\epsilon}):\) \(x\in R\), \(\epsilon >0\}\), where \(K_{x,\epsilon}=\{z\in C_+:\) \(| x-z| <\epsilon \}.\)
Theorem 4. Let \((B^ s_ p)_ A\) \((1\leq p<+\infty\), \(s\in R)\) be the space of functions f, analytic on the unit disk D and such that \[ \int^{2\pi}_{0}\int^{1}_{0}(1-r)^{k-s-1}| f^{(k)}(re^{it})|^ pr\quad drdt<+\infty, \] where k is any nonnegative integer greater than s. Let \(\psi\) be an analytic function on D with \(\psi\) (0)\(\neq 0\), whose Taylor coefficients are denoted by \({\hat \psi}\)(n). The operator \(\psi^*\) is then defined by \((\psi *f)(z)=\sum_{n\geq 0}{\hat \psi}(n)\hat f(n)z^ n\) for any analytic function f on D. Then (a) if \(\psi^*\) defines a continuous map from \((B^ s_ p)_ A\) into \(L^ 1(D,\mu)\), then \[ \mu (D)+\sup_{\lambda \in D}\int_{D}(1-| \lambda |)^{k+s}| \psi^{(k)}(\lambda z)| d\mu \quad (z)\quad (=c_ 1(\psi,\mu,k))<+\infty \] for any k with \(k\in N\), \(s+k>0\). (b) If \(c_ 1(\psi,\mu,k)<+\infty\) for some \(k\in N\) with \(s+k>0\), then \(\psi^*((B^ s_ 1)_ A)\subset L^ 1(D,\mu)\).
Reviewer: M.Hasumi

MSC:

30D55 \(H^p\)-classes (MSC2000)
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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