| Date: | February 4, 2022 |
| Time: | 10:20am |
| Location: | Zoom |
| Speaker: | Loredana Lanzani, Syracuse University |
| Title: | THE CAUCHY–SZEGŐ PROJECTION AND ITS COMMUTATOR FOR DOMAINS IN C^n WITH MINIMAL SMOOTHNESS: OPTIMAL BOUNDS |
| Abstract: | Let D ⊂ C^n be a bounded, strongly pseudoconvex domain whose boundary bD satisfies the minimal regularity condition of class C^2. A 2017 result of Lanzani & E. M. Stein states that the Cauchy–Szegő projection S_ω maps L^p(bD,ω) to L^p(bD, ω) continuously for any 1 < p < ∞ whenever the reference measure ω is a bounded, positive continuous multiple of induced Lebesgue measure. Here we show that S_ω (defined with respect to any measure ω as above) satisfies explicit, optimal bounds in L^p(bD, Ω_p), for any 1 < p < ∞ and for any Ω_p in the maximal class of A_p-measures, that is Ω_p = ψ_pσ where ψ_p is a Muckenhoupt A_p-weight and σ is the induced Lebesgue measure. As an application, we characterize boundedness in L^p(bD, Ω_p) with explicit bounds, and compactness, of the commutator [b, S_ω] for any A_p-measure Ω_p, 1 < p < ∞. We next introduce the notion of holomorphic Hardy spaces for A_p-measures, and we characterize boundedness and compactness in L^2(bD, Ω_2) of the commutator [b, S_{Ω_2}] where S_{Ω_2} is the Cauchy–Szegő projection defined with respect to any given A_2-measure Ω_2. Earlier results rely upon an asymptotic expansion and subsequent pointwise estimates of the Cauchy–Szegő kernel, but these are unavailable in our setting of minimal regularity of bD; at the same time, recent techniques that allow to handle domains with minimal regularity, are not applicable to A_p-measures. It turns out that the method of quantitative extrapolation is an appropriate replacement for the missing tools.
This is joint work with Xuan Thinh Duong (Macquarie University), Ji Li (Macquarie University) and Brett Wick (Washington University in St. Louis). |
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| Date: | March 23, 2022 |
| Time: | 4:00pm |
| Location: | Zoom |
| Speaker: | Xin Dong, University of Connecticut, Storrs |
| Title: | Bergman-Calabi diastasis and Kähler metric of constant holomorphic sectional curvature |
| Abstract: | With Bun Wong at UC Riverside, we study bounded domains in C^n with the Bergman metric of constant holomorphic sectional curvature. We give equivalent conditions for the domains being biholomorphic to a ball in terms of the exhaustiveness of the Bergman-Calabi diastasis. In particular, we prove that such domains are Lu Qi-Keng. We also extend a theorem of Lu towards the incomplete situation and characterize pseudoconvex domains that are biholomorphic to a ball possibly less a relatively closed pluripolar set. |
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| Date: | April 1, 2022 |
| Time: | 10:20am |
| Location: | ZOOM |
| Speaker: | Gian Maria Dall'Ara and Samuele Mongodi, Scuola Normale Superiore and Politenico di Milano |
| Title: | The Levi core: basic properties and applications |
| Abstract: | In the first part of this talk, Samuele will present briefly the construction of the core of the Levi distribution, highlighting the links with other known geometric invariants of boundaries of pseudoconvex domains, like finite-type points, local maximum sets, Levi currents. He will also sketch a parallel between the sequence of derived distributions from the Levi distribution to the core and Kohn's algorithm of multipliers' ideals.
In the second half of the talk, Gian Maria will discuss two ways in which the Levi core can be applied to problems in SCV, specifically he will address the question of whether the DF index of a domain is 1, the exact regularity of the d-bar Neumann problem, and subelliptic estimates. The subellipticity part is yet unpublished. |
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| Date: | April 21, 2022 |
| Time: | 1:00pm |
| Location: | Zoom |
| Speaker: | Ziming Shi, Rutgers University |
| Title: | Existence of Solutions for DBAR-Equation in Sobolev Spaces of Negative Index |
| Abstract: | Let Omega be a strictly pseudoconvex domain in C^n with C^{k+2} boundary, k \geq 1. We construct a DBAR-solution operator (depending on k) that gains 1/2 derivative in the Sobolev space H^{s,p}(Omega) for any 1 < p <\infty and s > 1/p -k$. If the domain is C^{\infty}, then there exists a DBAR-solution operator that gains 1/2 derivative in H^{s,p}(Omega) for all $s \in \mathbb{R}$. We obtain our solution operators through the method of homotopy formula; a new feature is the construction of "anti-derivative operators" on distributions defined on bounded Lipschitz domains. This is joint work with Liding Yao. |
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