Holomorphic Continuation via Laplace-Fourier series

关于一类具有平移不变性的非局部算子的唯一持续性原理

On the Unique Continuation Principle for a Class of Translation Invariant Nonlocal Operators

本文研究了Lévy算子的唯一持续性原理，建立了其与解析算子的关系，并通过新方法证明了分数拉普拉斯算子和某些函数的唯一持续性原理。

The heat equation with rough boundary conditions and holomorphic functional calculus

In this paper we consider the Laplace operator with Dirichlet boundary conditions on a smooth domain. We prove that it has a bounded $H^\infty$-calculus on weighted $L^p$-spaces for power weights which fall outside the classical class of $A_p$-weights. Furthermore, we characterize the domain of the operator and derive several consequences on elliptic and parabolic regularity. In particular, we obtain a new maximal regularity result for the heat equation with rough inhomogeneous boundary data.

On a Space of Infinitely Differentiable Functions on an Unbounded Convex Set in ${\mathbb R}^n$ Admitting Holomorphic Extension in ${\mathbb C}^n$ and its Dual

We consider a space of infinitely smooth functions on an unbounded closed convex set in ${\mathbb R}^n$. It is shown that each function of this space can be extended to an entire function in ${\mathbb C}^n$ satisfying some prescribed growth condition. Description of linear continuous functionals on this space in terms of their Fourier-Laplace transform is obtained. Also a variant of the Paley-Wiener-Schwartz theorem for tempered distributions is given it the paper.

Analytic Continuation of Resolvent Kernels on noncompact Symmetric Spaces

Let X=G/K be a symmetric space of noncompact type and let L be the Laplacian associated with a G-invariant metric on X. We show that the resolvent kernel of L admits a holomorphic extension to a Riemann surface depending on the rank of the symmetric space. This Riemann surface is a branched cover of the complex plane with a certain part of the real axis removed. It has a branching point at the bottom of the spectrum of L. It is further shown that this branching point is quadratic if the rank of X is odd, and is logarithmic otherwise. In case G has only one conjugacy class of Cartan subalgebras…

Space-Time Analyticity of Weak Solutions to Semilinear Parabolic Systems with Variable Coefficients

Analytic smooth solutions of a general, strongly parabolic semi-linear Cauchy problem of $2m$-th order in $\mathbb{R}^N\times (0,T)$ with analytic coefficients (in space and time variables) and analytic initial data (in space variables) are investigated. They are expressed in terms of holomorphic continuation of global (weak) solutions to the system valued in a suitable Besov interpolation space of $B^{s;p,p}$-type at every time moment $t\in [0,T]$. Given $0 < T'< T\leq \infty$, it is proved that any $B^{s;p,p}$-type solution $u\colon \mathbb{R}^N\times (0,T)\to \mathbb{C}^M$ with analytic ini…

圆盘与球面上不变拉普拉斯算子的谱理论——一种复分析方法

Spectral theory of the invariant Laplacian on the disk and the sphere -- a complex analysis approach

本文通过复分析方法，研究复化单位圆补集上不变拉普拉斯算子的全纯谱理论，给出本征空间闭Möbius不变子空间的内在分类，统一了双曲与球面拉普拉斯算子的本征值理论。