arXiv subjects

Andrew Raich

Publications and source records attributed to Andrew Raich.

33 records · Page 2Linked to original sources

arXiv2012

The Kerzman-Stein operator for piecewise continuously differentiable regions

The Kerzman-Stein operator is the skew-hermitian part of the Cauchy operator defined with respect to an unweighted hermitian inner product on a rectifiable curve. If the curve is continuously differentiable, the Kerzman-Stein operator is compact on the Hilbert space of square integrable functions; when there is a corner, the operator is noncompact. Here we give a complete description of the spectrum for a finite symmetric wedge and we show how this reveals the essential spectrum for curves that are piecewise continuously differentiable. We also give an explicit construction for a smooth curve whose Kerzman-Stein operator has large norm, and we demonstrate the variation in norm with respect to a continuously differentiable perturbation.

math.CV↗

arXiv2011

Defining Functions for Unbounded $C^m$ Domains

For a domain $Ω\subset\mathbb R^n$, we introduce the concept of a uniformly $C^m$ defining function. We characterize uniformly $C^m$ defining functions in terms of the signed distance function for the boundary and provide a large class of examples of unbounded domains with uniformly $C^m$ defining functions. Some of our results extend results from the bounded case.

math.DG↗

arXiv2011

Fundamental Solutions to $\Box_b$ on Certain Quadrics

The purpose of this article is to expand the number of examples for which the complex Green operator, that is, the fundamental solution to the Kohn Laplacian, can be computed. We use the Lie group structure of quadric submanifolds of $\mathbb C^n\times\mathbb C^m$ and the group Fourier transform to reduce the $\Box_b$ equation to ones that can be solved using modified Hermite functions. We use Mehler's formula and investigate 1) quadric hypersurfaces, where the eigenvalues of the Levi form are not identical (including possibly zero eigenvalues), and 2) the canonical quadrics in $\mathbb C^4$ of codimension two.

math.CV↗

arXiv2011

Green's function asymptotics near the internal edges of spectra of periodic elliptic operators. Spectral edge case

Precise asymptotics known for the Green's function of the Laplace operator have found their analogs for periodic elliptic operators of the second order at and below the bottom of the spectrum. Due to the band-gap structure of the spectra of such operators, the question arises whether similar results can be obtained near or at the edges of spectral gaps. As the result of this work shows, this is possible at a spectral edge in dimensions d>2.

math-ph↗

arXiv2011

Closed Range for $\bar\partial$ and $\bar\partial_b$ on Bounded Hypersurfaces in Stein Manifolds

We define weak $Z(q)$, a generalization of $Z(q)$ on bounded domains $Ω$ in a Stein manifold $M^n$ that suffices to prove closed range of $\bar\partial$. Under the hypothesis of weak $Z(q)$, we also show (i) that harmonic $(0,q)$-forms are trivial and (ii) if $\partialΩ$ satisfies weak $Z(q)$ and weak $Z(n-1-q)$, then $\dbar_b$ has closed range on $(0,q)$-forms on $\partialΩ$. We provide examples to show that our condition contains examples that are excluded from $(q-1)$-pseudoconvexity and the authors' previous notion of weak $Z(q)$.

math.CV↗

arXiv2010

An Aronsson type approach to extremal quasiconformal mappings

We study $C^2$ extremal quasiconformal mappings in space and establish necessary and sufficient conditions for a `localized' form of extremality in the spirit of the work of G. Aronsson on absolutely minimizing Lipschitz extensions. We also prove short time existence for smooth solutions of a gradient flow of QC diffeomorphisms associated to the extremal problem.

math.AP↗

arXiv2010

Heat Kernels, Smoothness Estimates and Exponential Decay

In this article, we establish Gaussian decay for the Box_b-heat kernel on polynomial models in C^2. Our technique attains the exponential decay via a partial Fourier transform. On the transform side, the problem becomes finding quantitative smoothness estimates on a heat kernel associated to the weighted dbar-operator on L^2(C). The bounds are established with Duhamel's formula and careful estimation.

math.CV↗

arXiv2009

Regularity results for $\bar\partial_b$ on CR-manifolds of hypersurface type

We introduce a class of embedded CR manifolds satisfying a geometric condition that we call weak $Y(q)$. For such manifolds, we show that dbar-b has closed range on $L^2$ and that the complex Green operator is continuous on $L^2$. Our methods involves building a weighted norm from a microlocal decomposition. We also prove that at any Sobolev level there is a weight such that the complex Green operator inverting the weighted Kohn Laplacian is continuous. Thus, we can solve the dbar-b equation in $C^\infty$.

math.CV↗

arXiv2009

Heat Equations and the Weighted $\bar\partial$-Problem

The purpose of this article is to establish regularity and pointwise upper bounds for the (relative) fundamental solution of the heat equation associated to the weighted dbar-operator in $L^2(C^n)$ for a certain class of weights. The weights depend on a parameter, and we find pointwise bounds for heat kernel, as well as its derivatives in time, space, and the parameter. We also prove cancellation conditions for the heat semigroup. We reduce the $n$-dimensional case to the one-dimensional case, and the estimates in one-dimensional case are achieved by Duhamel's principle and commutator properties of the operators. As an application, we recover estimates of heat kernels on polynomial models in $C^2$.

math.AP↗

arXiv2009

The $\Box_b$-heat equation on quadric manifolds

In this article, we give an explicit calculation of the partial Fourier transform of the $\Box_b$-heat equation on quadric submanifolds of $M\subset C^n\times C^m$. As a consequence, we can also compute the heat kernel associated to the weighted dbar-equation in $C^n$ when the weight is given by $\exp(-ϕ(z,z)\cdotλ)$ where $ϕ: C^n\times C^n\to C^m$ is a quadratic, sesquilinear form and $λ\in R^m$. Our method involves the representation theory of the Lie group $M$ and the group Fourier transform.

math.CV↗

arXiv2008

Compactness of the Complex Green Operator on CR-Manifolds of Hypersurface Type

The purpose of this article is to study compactness of the complex Green operator on CR manifolds of hypersurface type. We introduce (CR-P_q), a potential theoretic condition on $(0,q)$-forms that generalizes Catlin's property (P_q) to CR manifolds of arbitrary codimension. We prove that if an embedded CR-manifold of hypersurface type satisfies (CR-P_q) and (CR-P_{n-1-q}) and is of real dimension at least five, then the complex Green operator is a compact operator on the Sobolev spaces $H^s_{0,q}(M)$, if $1\leq q \leq n-2$ and $s\geq 0$. We use CR-plurisubharmonic functions to build a microlocal norm that controls the totally real direction of the tangent bundle.

math.CV↗

arXiv2007

A Simplified Calculation for the Fundamental Solution to the Heat Equation on the Heisenberg Group

Let $L = -1/4 (\sum_{j=1}^n(X_j^2+Y_j^2)+iγT)$ where $γ$ is a complex number, $X_j$, $Y_j$, and $T$ are the left invariant vector fields of the Heisenberg group structure for $R^n \times R^n \times R$. We explicitly compute the Fourier transform (in the spatial variables) of the fundamental solution of the Heat Equation $\partial_sρ= -Lρ$. As a consequence, we have a simplified computation of the Fourier transform of the fundamental solution of the $\Box_b$-heat equation on the Heisenberg group and an explicit kernel of the heat equation associated to the weighted dbar-operator in $C^n$ with weight $\exp(-τP(z_1,...,z_n))$ where $P(z_1,...,z_n) = 1/2(x_1^2 + >... x_n^2)$, $z_j=x_j+iy_j$, and $τ\in R$.

math.AP↗

arXiv2006

One-Parameter Families of Operators in $\mathbb{C}$

We develop classes of one-parameter families (OPF) of operators on $C^\infty_c(\mathbb{C})$ which characterize the behavior of operators associated to the $\bar\partial$-problem in $L^2(\mathbb{C},e^{-2p})$ where $p$ is a subharmonic, nonharmonic polynomial. We prove that an order 0 OPF operator extends to a bounded operator from $L^q(\mathbb{C})$ to itself, $1<q<\infty$, with a bound that depends on $q$ and the degree of $p$ but not on the parameter $τ$ or the coefficients of $p$. Last, we show that there is a one-to-one correspondence given by the partial Fourier transform in $τ$ between OPF operators of order $m\leq 2$ and nonisotropic smoothing (NIS) operators of order $m\leq 2$ on polynomial models in $\mathbb{C}^2$.

math.CV↗

arXiv2006

Heat Equations in $\mathbb{R}\times\mathbb{C}$

Let $p:\mathbb{C}\to\mathbb{R}$ be a subharmonic, nonharmonic polynomial and $τ$ a real parameter. Define $\bar{Z}_{τp} = \partial_{\bar z} + τp_{\bar z}$, a closed, densely-defined operator on $L^2(\mathbb{C})$. If $\Box_{τp} = \bar{Z}_{τp}\bar{Z}_{τp}^*$ and $τ>0$, we solve the heat equation $ (\partial_s + \Box_{τp}) u =0$, $u(0,z) = f(z)$, on $(0,\infty)\times\mathbb{C}$. The solution comes via the heat semigroup $e^{-s\Box_{τp}}$, and we show that $u(s,z)$ is given as integration of the intial condition against a distributional kernel $H_{τp}(s,z,w)$. We prove that $H_{τp}$ is $C^\infty$ off the diagonal $\{(s,z,w):s=0 \text{and }z=w\}$ and that $H_{τp}$ and its derivatives have exponential decay.

math.CV↗

arXiv2006

Pointwise Estimates for Relative Fundamental Solutions of Heat Equations in $\mathbb{R}\times\mathbb{C}$

Let $p:C\to R$ be a subharmonic, nonharmonic polynomial and $τ\in R$ a parameter. Define $\bar Z_{τp} = \partial_{\bar z} + τp_{\bar z} = e^{-τp} p_{\bar z} e^{τp}$, a closed, densely defined operator on $L^2(C)$. If $\Box_{τp} = \bar Z_{τp}\bar Z^*_{τp}$ and $\tilde\Box_{τp} = \bar Z^*_{τp}\bar Z_{τp}$, we solve the heat equations $\partial_s u + \Box_{τp} u=0$, $u(0,z)=f(z)$ and $\partial_s \tilde u + \tilde\Box_{τp} \tilde u=0$, $\tilde u(0,z) = \tilde f(z)$. We write the solutions via heat semigroups and show that the solutions can be written as integrals against distributional kernels. We prove that the kernels are $C^\infty$ off of the diagonal $\{(s,z,w) : s=0 \text{and} z=w\}$ and find pointwise bounds for the kernels and their derivatives.

math.CV↗