$L^2$ estimate for polynomials of the Laplace operator with Gaussian measure

Shaoyu Dai; Yang Liu; Yifei Pan

$L^2$ estimate for polynomials of the Laplace operator with Gaussian measure

Analysis of PDEs 2021-06-09 v1 Complex Variables

Authors: Shaoyu Dai , Yang Liu , Yifei Pan

Abstract

Let $P(\Delta)$ be a polynomial of the Laplace operator $\Delta=\sum_{j=1}^n\frac{\partial^2}{\partial x^2_j}$ on $\mathbb{R}^n$ . We prove the existence of weak solutions of the equation $P(\Delta)u=f$ and the existence of a bounded right inverse of the differential operator $P(\Delta)$ in the weighted Hilbert space with Gaussian measure, i.e., $L^2(\mathbb{R}^n,e^{-|x|^2})$ .

Keywords

laplace operator gaussian measure l2 estimate weak solution bounded right inverse weighted hilbert space

Cite

@article{arxiv.2106.03938,
  title  = {$L^2$ estimate for polynomials of the Laplace operator with Gaussian measure},
  author = {Shaoyu Dai and Yang Liu and Yifei Pan},
  journal= {arXiv preprint arXiv:2106.03938},
  year   = {2021}
}

Comments

arXiv admin note: text overlap with arXiv:1909.12477

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