arXiv subjects
Publications and source records attributed to Shaoyu Dai.
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., $L^2(\mathbb{R}^n,e^{-|x|^2})$.
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.
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.
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.
In this paper we prove a Schwarz lemma for harmonic mappings between the unit balls in real Euclidean spaces. Roughly speaking, our result says that under a harmonic mapping between the unit balls in real Euclidean spaces, the image of a smaller ball centered at origin can be controlled. This extends the related result proved by Chen in complex plane.
We prove a high order Schwarz-Pick lemma for mappings between unit balls in complex spaces in terms of the Bergman metric. From this lemma, Schwarz-Pick estimates for partial derivatives of arbitrary order of mappings are deduced.