Abstract
Let \(A^p_\omega \) denote the Bergman space in the unit disc induced by a radial weight \(\omega \) with the doubling property \(\int _{r}^1\omega (s)\,ds\le C\int _{\frac{1+r}{2}}^1\omega (s)\,ds\). The positive Borel measures such that the differentiation operator of order \(n\in \mathbb {N}\cup \{0\}\) is bounded from \(A^p_\omega \) into \(L^q(\mu )\) are characterized in terms of geometric conditions when \(0<p,q<\infty \). En route to the proof a theory of tent spaces for weighted Bergman spaces is built.
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Acknowledgments
The authors wish the thank Brett Wick for pointing out the reference [16] with regard to maximal functions.
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This research was supported in part by the Ramón y Cajal program of MICINN (Spain); by Ministerio de Educación y Ciencia, Spain, projects MTM2011-25502 and MTM2011-26538; by La Junta de Andalucía, (FQM210) and (P09-FQM-4468); by Academy of Finland project no. 268009, by Väisälä Foundation of Finnish Academy of Science and Letters, and by Faculty of Science and Forestry of University of Eastern Finland project no. 930349.
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Peláez, J.Á., Rättyä, J. Embedding theorems for Bergman spaces via harmonic analysis. Math. Ann. 362, 205–239 (2015). https://doi.org/10.1007/s00208-014-1108-5
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DOI: https://doi.org/10.1007/s00208-014-1108-5