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Register a new user$mathcal{H}^p$-corona problem and convex domains of finite type
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Authors
Willliam Alexandre
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We prove that the $mathcal{H}^p$-corona problem has a solution for convex domains of finite type in $mathbb{C}^n$, $n ge 2$.
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In this paper, we introduce a dyadic structure on convex domains of finite type via the so-called dyadic flow tents. This dyadic structure allows us to establish weighted norm estimates for the Bergman projection $P$ on such domains with respect to Muckenhoupt weights. In particular, this result gives an alternative proof of the $L^p$ boundedness of $P$. Moreover, using extrapolation, we are also able to derive weighted vector-valued estimates and weighted modular inequalities for the Bergman projection.
Let $Dsubset mathbb C^n$ be a bounded domain. A pair of distinct boundary points ${p,q}$ of $D$ has the visibility property provided there exist a compact subset $K_{p,q}subset D$ and open neighborhoods $U_p$ of $p$ and $U_q$ of $q$, such that the real geodesics for the Kobayashi metric of $D$ which join points in $U_p$ and $U_q$ intersect $K_{p,q}$. Every Gromov hyperbolic convex domain enjoys the visibility property for any couple of boundary points. The Goldilocks domains introduced by Bharali and Zimmer and the log-type domains of Liu and Wang also enjoy the visibility property. In this paper we relate the growth of the Kobayashi distance near the boundary with visibility and provide new families of convex domains where that property holds. We use the same methods to provide refinements of localization results for the Kobayashi distance, and give a localized sufficient condition for visibility. We also exploit visibility to study the boundary behavior of biholomorphic maps.
In the late ten years, the resolution of the equation $barpartial u=f$ with sharp estimates has been intensively studied for convex domains of finite type by many authors. In this paper, we consider the case of lineally convex domains. As the method used to obtain global estimates for a support function cannot be carried out in this case, we use a kernel that does not gives directly a solution of the $barpartial$-equation but only a representation formula which allows us to end the resolution of the equation using Kohns $L^2$ theory. As an application we give the characterization of the zero sets of the functions of the Nevanlinna class for lineally convex domains of finite type.
We give a necessary complex geometric condition for a bounded smooth convex domain in Cn, endowed with the Kobayashi distance, to be Gromov hyperbolic. More precisely, we prove that if a smooth bounded convex domain contains an analytic disk in its boundary, then the domain is not Gromov hyperbolic for the Kobayashi distance. We also provide examples of bounded smooth convex domains that are not strongly pseudoconvex but are Gromov hyperbolic.
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 $mathcal{Q}_p$.
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