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Register a new userA survey on reverse Carleson measures
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This is a survey on reverse Carleson measures for various Hilbert spaces of analytic functions. These spaces include Hardy, Bergman, certain harmonically weighted Dirichlet, Paley-Wiener, Fock, model, and de Branges-Rovnyak spaces.
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We investigate a variant of the Beurling-Ahlfors extension of quasisymmetric homeomorphisms of the real line that is given by the convolution of the heat kernel, and prove that the complex dilatation of such a quasiconformal extension of a strongly symmetric homeomorphism (i.e. its derivative is an ${rm A}_infty$-weight whose logarithm is in VMO) induces a vanishing Carleson measure on the upper half-plane.
The purposes of this paper are two fold. First, we extend the method of non-homogeneous harmonic analysis of Nazarov, Treil and Volberg to handle Bergman--type singular integral operators. The canonical example of such an operator is the Beurling transform on the unit disc. Second, we use the methods developed in this paper to settle the important open question about characterizing the Carleson measures for the Besov--Sobolev space of analytic functions $B^sigma_2$ on the complex ball of $mathbb{C}^d$. In particular, we demonstrate that for any $sigma> 0$, the Carleson measures for the space are characterized by a T1 Condition. The method of proof of these results is an extension and another application of the work originated by Nazarov, Treil and the first author.
We completely characterize those positive Borel measures $mu$ on the unit ball $mathbb{B}_ n$ such that the Carleson embedding from Hardy spaces $H^p$ into the tent-type spaces $T^q_ s(mu)$ is bounded, for all possible values of $0<p,q,s<infty$.
Let $mu$ be a nonnegative Borel measure on the open unit disk $mathbb{D}subsetmathbb{C}$. This note shows how to decide that the Mobius invariant space $mathcal{Q}_p$, covering $mathcal{BMOA}$ and $mathcal{B}$, is boundedly (resp., compactly) embedded in the quadratic tent-type space $T^infty_p(mu)$. Interestingly, the embedding result can be used to determine the boundedness (resp., the compactness) of the Volterra-type and multiplication operators on $mathcal{Q}_p$.
In this paper, we study the behavior of the weighted composition operators acting on Bergman spaces defined on strictly pseudoconvex domains via the sparse domination technique from harmonic analysis. As a byproduct, we also prove a weighted type estimate for the weighted composition operators which is adapted to Sawyer-testing conditions. Our results extend the work by the first author, Li, Shi and Wick under a much more general setting.
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