Embedding theorems for Bergman spaces via harmonic analysis

By J. A. Pel'aez, Jouni Rattya. 2014

96 citations recorded by the source corpus.

Let $$A^p_\omega $$Aωp denote the Bergman space in the unit disc induced by a radial weight $$\omega $$ω with the doubling property $$\int _{r}^1\omega (s)\,ds\le C\int _{\frac{1+r}{2}}^1\omega (s)\,ds$$∫r1ω(s)ds≤C∫1+r21ω(s)ds. The positive Borel measures such that the differentiation operator of order $$n\in \mathbb {N}\cup \{0\}$$n∈N∪{0} is bounded from $$A^p_\omega $$Aωp into $$L^q(\mu )$$Lq(μ) are characterized in terms of geometric conditions when $$0<p,q<\infty $$0<p,q<∞. En route to the proof a theory of tent spaces for weighted Bergman spaces is built.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/274c0c23961334b55036eafab6f774de5df0c172

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In Applied Mathematics

1. J. A. Pel'aez
2. Jouni Rattya
3. Jordi Pau
4. Kian Sierra
5. A. Perälä
6. S. Miihkinen
7. Maofa Wang
8. O. Constantin

Citations in the corpus

• Small weighted Bergman spaces (2015, methodology): “Plenty of results in [49, 50, 51] show that spaces Aω induced by rapidly increasing weights (Section 2 below for a definition), lie “closer”to H than any Aα.”
• Trace class criteria for Toeplitz and composition operators on small Bergman spaces (2014, methodology): “own that for such ω, many finer function theoretic properties of Ap ω are very different from those of A p α and, in particular, its harmonic analysis is similar to that of Hardy spaces in many aspects [17, 18, 19]. For any ω ∈ Db, the norm convergence in A2 ω implies the uniform convergence on compac”
• Embedding Bergman spaces into tent spaces (2015, background): “The recent studies [14,15] show that tent spaces have natural analogues for Bergman spaces, and they may play a role in the theory of weighted Bergman spaces similar to that of the original tent spaces in the Hardy space case.”
• Volterra type integration operators from Bergman spaces to Hardy spaces (2019, background): “that can be proved in the same manner as some variants appearing in [11, 30] (see also [27]).”