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Register a new userInterpolating sequences for the Bergman space and the $barpartial$-equation in weighted $L^p$
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Authors
Daniel H. Luecking
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The author showed that a sequence in the unit disk is a zero sequence for the Bergman space $A^p$ if and only if a certain weighted space $L^p(W}$ contains a nontrivial analytic function. In this paper it is shown that the sequence is an interpolating sequence for $A^p$ if and only if it is separated in the hyperbolic metric and the $barpartial$-equation $(1 - |z|^2)barpartial u = f$ has a solution $u$ belonging to $L^p(W)$ for every $f$ in $L^p(W)$.
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We obtain sharp ranges of $L^p$-boundedness for domains in a wide class of Reinhardt domains representable as sub-level sets of monomials, by expressing them as quotients of simpler domains. We prove a general transformation law relating $L^p$-boundedness on a domain and its quotient by a finite group. The range of $p$ for which the Bergman projection is $L^p$-bounded on our class of Reinhardt domains is found to shrink as the complexity of the domain increases.
A class of pseudoconvex domains in $mathbb{C}^{n}$ generalizing the Hartogs triangle is considered. The $L^p$ boundedness of the Bergman projection associated to these domains is established, for a restricted range of $p$ depending on the fatness of domains. This range of $p$ is shown to be sharp.
Regularity and irregularity of the Bergman projection on $L^p$ spaces is established on a natural family of bounded, pseudoconvex domains. The family is parameterized by a real variable $gamma$. A surprising consequence of the analysis is that, whenever $gamma$ is irrational, the Bergman projection is bounded only for $p=2$.
We study almost sure separating and interpolating properties of random sequences in the polydisc and the unit ball. In the unit ball, we obtain the 0-1 Komolgorov law for a sequence to be interpolating almost surely for all the Besov-Sobolev spaces $B_{2}^{sigma}left(mathbb{B}_{d}right)$, in the range $0 < sigmaleq1 / 2$. For those spaces, such interpolating sequences coincide with interpolating sequences for their multiplier algebras, thanks to the Pick property. This is not the case for the Hardy space $mathrm{H}^2(mathbb{D}^d)$ and its multiplier algebra $mathrm{H}^infty(mathbb{D}^d)$: in the polydisc, we obtain a sufficient and a necessary condition for a sequence to be $mathrm{H}^infty(mathbb{D}^d)$-interpolating almost surely. Those two conditions do not coincide, due to the fact that the deterministic starting point is less descriptive of interpolating sequences than its counterpart for the unit ball. On the other hand, we give the $0-1$ law for random interpolating sequences for $mathrm{H}^2(mathbb{D}^d)$.
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