Weighted Bergman spaces induced by rapidly incresing weights
By J. A. Pel'aez, Jouni Rattya. 2012
198 citations recorded by the source corpus.
This monograph is devoted to the study of the weighted Bergman space $A^p_\om$ of the unit disc $\D$ that is induced by a radial continuous weight $\om$ satisfying {equation}\label{absteq} \lim_{r\to 1^-}\frac{\int_r^1\om(s)\,ds}{\om(r)(1-r)}=\infty.\tag{\dag} {equation} Every such $A^p_\om$ lies between the Hardy space $H^p$ and every classical weighted Bergman space $A^p_\a$. Even if it is well known that $H^p$ is the limit of $A^p_\a$, as $\a\to-1$, in many respects, it is shown that $A^p_\om$ lies "closer" to $H^p$ than any $A^p_\a$, and that several finer function-theoretic properties of $A^p_\a$ do not carry over to $A^p_\om$. As to concrete objects to be studied, positive Borel measures $\mu$ on $\D$ such that $A^p_\om\subset L^q(\mu)$, $0<p\le q<\infty $, are characterized in terms of a neat geometric condition involving Carleson squares. It is also proved that each $f\in A^p_\om$ can be represented in the form $f=f_1\cdot f_2$, where $f_1\in A^{p_1}_\om$, $f_2\in A^{p_2}_\om$ and $\frac{1}{p_1}+ \frac{1}{p_2}=\frac{1}{p}$. Because of the tricky nature of $A^p_\om$ several new concepts are introduced. It gives raise to a some what new approach to the study of the integral operator $$ T_g(f)(z)=\int_{0}^{z}f(\zeta)\,g'(\zeta)\,d\zeta. $$ This study reveals the fact that $T_g:A^p_\om\to A^p_\om$ is bounded if and only if $g$ belongs to a certain space of analytic functions that is not conformally invariant. The symbols $g$ for which $T_g$ belongs to the Schatten $p$-class $\SSS_p(A^2_\om)$ are also described. Furthermore, techniques developed are applied to the study of the growth and the oscillation of analytic solutions of (linear) differential equations.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/5518bdb85eed0ad2a14939f0dbdfd5706aa54060
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Citations in the corpus
- Embedding theorems for Bergman spaces via harmonic analysis (2014, background): “To some extent this is natural because a weighted Bergman space Aω induced by ω ∈ D̂ may lie essentially much closer to the Hardy space Hp than any standard Bergman space Aα [14].”
- Generalized Hilbert operators on weighted Bergman spaces (2012, background): “The reader is invited to see [20] for more information on this topic.”
- Two weight inequality for Bergman projection (2014, unknown): “Note that in [20] the space Hα(ω ⋆) is defined by the condition ∞ ∑”
- Small weighted Bergman spaces (2015, background): “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.”
- Integration operators between Hardy spaces on the unit ball of $\Cn$ (2013, background): “ 7] describing the boundedness and compactness of the operator Jg in Hardy and Bergman spaces, the mentioned operator became extremely popular, being studied in many spaces of analytic functions (see [4, 5, 6, 7, 13, 27, 28] for example). As far as we know, the generalization of the operator Jg acti”
- Integral Operators, Embedding Theorems and a Littlewood–Paley Formula on Weighted Fock Spaces (2013, background): “[3, 6] for Hardy spaces, [5, 19, 23] for weighted Bergman spaces, or the survey [2] and the references therein).”
- Trace class criteria for Toeplitz and composition operators on small Bergman spaces (2014, result): “Note that in [17] the space Hα(ω ⋆) is defined by the condition ∞ ∑”