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40 pages, 463 KB

Open AccessArticle

by Elisabetta Barletta, Sorin Dragomir and Francesco Esposito

Mathematics 2022, 10(3), 371; https://doi.org/10.3390/math10030371 - 25 Jan 2022

Viewed by 2363

Beltrami equations

{\bar{L}}_{t} (g) = μ (\cdot, t) L_{t} (g)

S^{3}

(where

L_{t}

| t | < 1

, are the Rossi operators i.e.,

L_{t}

spans the globally [...] Read more.

Beltrami equations

{\bar{L}}_{t} (g) = μ (\cdot, t) L_{t} (g)

S^{3}

(where

L_{t}

| t | < 1

, are the Rossi operators i.e.,

L_{t}

spans the globally nonembeddable CR structure

H (t)

S^{3}

discovered by H. Rossi) are derived such that to describe quasiconformal mappings

f : S^{3} \to N \subset C^{2}

from the Rossi sphere

(S^{3}, H (t))

. Using the Greiner–Kohn–Stein solution to the Lewy equation and the Bargmann representations of the Heisenberg group, we solve the Beltrami equations for Sobolev-type solutions

g_{t}

such that

g_{t} - v \in W_{F}^{1, 2} (S^{3}, θ)

with

v \in {CR}^{\infty} (S^{3}, H (0))

. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications Part II)

42 pages, 569 KB

Open AccessReview

Weighted Bergman Kernels and Mathematical Physics

by Elisabetta Barletta, Sorin Dragomir and Francesco Esposito

Axioms 2020, 9(2), 48; https://doi.org/10.3390/axioms9020048 - 29 Apr 2020

Cited by 1 | Viewed by 4164

Abstract

We review several results in the theory of weighted Bergman kernels. Weighted Bergman kernels generalize ordinary Bergman kernels of domains

Ω \subset C^{n}

but also appear locally in the attempt to quantize classical states of mechanical systems whose classical phase space is [...] Read more.

We review several results in the theory of weighted Bergman kernels. Weighted Bergman kernels generalize ordinary Bergman kernels of domains

Ω \subset C^{n}

but also appear locally in the attempt to quantize classical states of mechanical systems whose classical phase space is a complex manifold, and turn out to be an efficient computational tool that is useful for the calculation of transition probability amplitudes from a classical state (identified to a coherent state) to another. We review the weighted version (for weights of the form

γ = | φ |^{m}

on strictly pseudoconvex domains

Ω = {φ < 0} \subset C^{n}

) of Fefferman’s asymptotic expansion of the Bergman kernel and discuss its possible extensions (to more general classes of weights) and implications, e.g., such as related to the construction and use of Fefferman’s metric (a Lorentzian metric on

\partial Ω \times S^{1}

). Several open problems are indicated throughout the survey. Full article

(This article belongs to the Special Issue Geometric Analysis and Mathematical Physics)

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