arXiv subjects
Publications and source records attributed to Yifei Pan.
We construct examples of twice differentiable functions in $\mathbb{R}^n$ with continuous Laplacian and unbounded Hessian. The same construction is also applicable to higher order differentiability.
We prove a general local existence theorem for nonlinear systems of Cauchy-Riemann operator of any order in one complex variable with initial values at a given point, which is a counterpart of local existence of ODE.
In this paper, we investigate the unique continuation property for the inequality $|\bar\partial u| \le V|u|$, where $u$ is a vector-valued function from a domain in $\mathbb C^n$ to $\mathbb C^N$, and the potential $V\in L^2$. We show that the strong unique continuation property holds when $n=1$, and the weak unique continuation property holds when $n\ge 2$. In both cases, the $L^2$ integrability condition on the potential is optimal.
Let $P(Δ)$ be a polynomial of the Laplace operator $Δ=\sum_{j=1}^n\frac{\partial^2}{\partial x^2_j}$ on $\mathbb{R}^n$. We prove the existence of weak solutions of the equation $P(Δ)u=f$ and the existence of a bounded right inverse of the differential operator $P(Δ)$ in the weighted Hilbert space with Gaussian measure, i.e., $L^2(\mathbb{R}^n,e^{-|x|^2})$.
The purpose of this paper is to study Hölder estimates for the $\bar\partial$ problem for $(p,q)$ forms on products of general planar domains. As indicated by an example of Stein and Kerzman, solutions to the $\bar\partial$ problem on product domains in $\mathbb C^n (n\ge 2)$ does not gain regularity in Hölder spaces. Making use of an integral representation of Nijenhuis and Woolf, we show that given a $\bar\partial$-closed $(p,q)$ form with $C^{k,α}$ components, $0\le p\le n, 1\le q\le n$, $k\in \mathbb Z^+\cup \{0\}, 0<α\le 1$, there is a $C^{k, α'}$ solution to the $\bar\partial$ problem on product domains for any $0<α'<α$ with the desired Hölder estimate.
Let $Ω\subset\mathbb{R}^n, n\geq 2$, be an open set. For an elliptic differential operator $L$ on $Ω$ with real analytic coefficients and a point $p\inΩ$, we construct a smooth function $g$ with the following properties: $g$ is flat at $p$ and the equation $Lu=g$ has no smooth local solution $u$ that is flat at $p$.
We study the minimal regularity required on the datum to guarantee the existence of classical $C^1$ solutions to the inhomogeneous Cauchy-Riemann equations on planar domains.
In this paper, we prove the existence of solutions of the Poincaré-Lelong equation $\sqrt{-1}\partial\bar{\partial}u=f$ on a strictly convex bounded domain $Ω\subset\mathbb{C}^n$ $(n\geq1)$, where $f$ is a $d$-closed $(1,1)$ form and is in the weighted Hilbert space $L^2_{(1,1)}(Ω,e^{-φ})$. The novelty of this paper is to apply a weighted $L^2$ version of Poincaré Lemma for real $2$-forms, and then apply Hörmander's $L^2$ solutions for Cauchy-Riemann equations.
This paper is motivated by a claim in the classical textbook of Muskhelishvili concerning the Cauchy singular integral operator $S$ on Hölder functions with parameters. To the contrary of the claim, a counter example was constructed by Tumanov which shows that $S$ with parameters fails to maintain the same Hölder regularity with respect to the parameters. In view of the example, the behavior of the Cauchy singular integral operator with parameters between a type of Log-Hölder spaces is investigated to obtain the sharp norm estimates. At the end of the paper, we discuss its application to the $\bar\partial$ problem on product domains.
In this paper, we prove the existence of (global) solutions of the Poincaré-Lelong equation $\partial\overline{\p}u=f$, where $f$ is a $d$-closed $(1,1)$ form and is in the weighted Hilbert space with Gaussian measure, i.e., $L^2_{(1,1)}(\mathbb{C}^n,e^{-|z|^2})$. The novelty of this paper is to apply a weighted $L^2$ version of Poincaré Lemma for $2$-forms, and then apply Hörmander's $L^2$ solutions for Cauchy-Riemann equations. In the both cases, the same weight $e^{-|z|^2}$ is used.
In this note, we prove the existence of weak solutions of a Poisson type equation in the weighted Hilbert space $L^2(\mathbb{R}^n,e^{-|x|^2})$.
Using Hörmander $L^2$ method for Cauchy-Riemann equations from complex analysis, we study a simple differential operator $\bar{\partial}^k+a$ of any order (densely defined and closed) in weighted Hilbert space $L^2(\mathbb{C},e^{-|z|^2})$ and prove the existence of a right inverse that is bounded.
Let $Ω\subset\mathbb{C}^n$ be a product of one-dimensional open bounded domains with $C^{1,α}$ boundary, where $0<α<1$. Using methods from complex analysis in one variable, we construct an integral operator that solves $\bar\partial$ in $Ω$ with supnorm estimates when the datum is in $C^{n-1,α}(Ω)$.
We establish elliptic regularity for nonlinear inhomogeneous Cauchy-Riemann equations under minimal assumptions, and give a counterexample in a borderline case. In some cases where the inhomogeneous term has a separable factorization, the solution set can be explicitly calculated. The methods also give local parametric formulas for pseudoholomorphic curves with respect to some continuous almost complex structures.
In this paper, we give a general boundary Schwarz lemma for holomorphic mappings between unit balls in any dimensions. It is proved that if the mapping $f\in C^{1+α}$ at $z_0\in \partial \mathbb B^n$ with $f(z_0)=w_0\in \partial \mathbb B^N$ for any $n,N\geq 1$, then the Jacobian matrix $J_f(z_0)$ maps the tangent space $T_{z_0}(\partial \mathbb B^n)$ to $T_{w_0}(\partial \mathbb B^N)$, and the holomorphic tangent space $T^{(1,0)}_{z_0}(\partial \mathbb B^n)$ to $T^{(1,0)}_{w_0}(\partial \mathbb B^N)$ as well.
In this paper, we prove a general Schwarz lemma at the boundary for holomorphic mappings from the polydisc to the unit ball in any dimensions. For the special case of one complex variable, the obtained results give the classic boundary Schwarz lemma.
In this paper we prove a Schwarz-Pick lemma for the modulus of holomorphic mappings from the polydisk into the unit ball. This result extends some related results.
In this paper we prove a Schwarz-Pick lemma for the modulus of holomorphic mappings between the unit balls in complex spaces. This extends the classical Schwarz-Pick lemma and the related result proved by Pavlovic.