Embedding theorems for Bergman spaces via harmonic analysis. (English) Zbl 1333.46032
In this paper, the authors characterize, in terms of geometric conditions, the positive Borel measures \(\mu \) on the unit disc \(\mathbb{D}\) such that
\[
\left( \int \left| f^{(n)}(z)\right| ^{q}d\mu (z)\right) ^{1/q}\leq C\left\| f\right\| _{A_{w}^{p}},
\]
where \(n\in \mathbb{N}\cup \{0\},\) \(A_{w}^{p}\) is the Bergman space induced by \(w\) a radial weights such that \(\hat{w}(r)=\int_{r}^{1}w(s)\,ds\) satisfies that \( \hat{w}(r)/\hat{w}(\frac{1+r}{2})\) is bounded and \(0<p,q<\infty\).
Reviewer: Joaquím Martín (Barcelona)
MSC:
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 42B35 | Function spaces arising in harmonic analysis |
| 30H20 | Bergman spaces and Fock spaces |
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