L^p-quasicontractiveness and Kernel estimates for semigroups generated by systems of elliptic operators

TL;DR

The paper develops $L^p$-quasicontractivity and kernel estimates for semigroups generated by strongly coupled elliptic operators with unbounded coefficients on $\Omega\subset\mathbb{R}^d$ acting on $\mathbb{C}^m$-valued functions. Using a vector-valued version of Nittka's criterion, it proves that the $L^2$-generated semigroup extends to a bounded analytic semigroup on $L^p(\Omega;\mathbb{C}^m)$ for $p$ in an explicit interval around 2, with operator norms controlled by the coefficient data. Under refined assumptions, the authors obtain generalized Gaussian kernel bounds with respect to the distance $d_{Q,V,\beta}$ that encodes growth of diffusion and potential at infinity, and in the classical bounded-coefficient setting recover standard Gaussian bounds and even $H^\infty$-calculus results. These results provide a robust framework for vector-valued elliptic systems with unbounded coefficients, including new kernel estimates for first-order coupled operators, broadening the spectral and parabolic theory in this setting.

Abstract

This paper focuses on systems of strongly coupled elliptic operators whose coefficients may be unbounded and are defined on a domain $Ω\subseteq \mathbb{R}^d$. It is shown that a quasi-contractive semigroup in $L^p$-spaces can be associated with such operators for values of $p$ belonging to an interval that contains $2$ as an interior point. Then, under refined assumptions and considering systems of elliptic operators coupled up to first order, new kernel estimates are established with respect to a distance function that accounts for the growth of the diffusion coefficients and the potential term at infinity.

Theorems & Definitions (26)

• Theorem 2.1
• Remark 2.2
• Theorem 2.3: Nittka's Criterion
• Remark 2.4
• Remark 2.6: Proposition A.1 & Corollary A.2 in AngLorMan
• Lemma 2.7
• proof
• Lemma 2.9
• proof
• Proposition 2.10