# Events for 02/04/2022 from all calendars

## Several Complex Variables Seminar

**Time: ** 10:20AM - 11:20AM

**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).

## Colloquium - Simone Cecchini

**Time: ** 3:30PM - 4:30PM

**Location: ** BLOC 117

**Speaker: **Simone Cecchini, University of Gottingen, Germany

**Description: **

**Title:** A long neck principle for Riemannian spin manifolds with positive scalar curvature

**Abstract:** We present results in index theory on compact Riemannian spin manifolds with boundary in the case when the topological information is encoded by bundles which are supported away from the boundary. As a first application, we establish a ``long neck principle'' for a compact Riemannian spin n-manifold with boundary X, stating that if scal(X) ≥ n(n-1) and there is a nonzero degree map f into the n-sphere which is area decreasing, then the distance between the support of the differential of f and the boundary of X is at most π/n. This answers, in the spin setting, a question asked by Gromov. As a second application, we consider the case of a Riemannian n-manifold V diffeomorphic to Nx [-1,1], where N is the (n-1)-torus or more in general a closed spin manifold with a suitable nonvanishing topological invariant. In this case, we show that, if scal(V) ≥ n(n-1), then the distance between the boundary components of V is at most 2π/n. This last constant is sharp by an argument due to Gromov.