$L^p$-bounds for semigroups generated by non-elliptic quadratic differential operators

Abstract

In this note, we establish $L^p$-bounds for the semigroup $e^{-t{q}^w(x,D)}$, $t\ge 0$, generated by a quadratic differential operator $q^w(x,D)$ on ${\mathbb{R}}^n$ that is the Weyl quantization of a complex-valued quadratic form $q$ defined on the phase space ${\mathbb{R}}^{2n}$ with non-negative real part $\mathrm{Re}q\ge 0$ and trivial singular space. Specifically, we show that $e^{-t{q}^w(x,D)}$ is bounded from $L^p({\mathbb{R}}^n)$ to $L^q({\mathbb{R}}^n)$ for all $t>0$ whenever $1\le p\le q\le \infty$, and we prove bounds on $\parallel {e^{-t{q}^w(x,D)}}_{L^p\to L^q}\parallel$ in both the large $t\gg 1$ and small $0<t\ll 1$ time regimes.Regardin g $L^p\to L^q$ bounds for the evolution semigroup at large times, we show that $\parallel {e^{-t{q}^w(x,D)}}_{L^p\to L^q}\parallel$ is exponentially decaying as $t\to \infty$, and we determine the precise rate of exponential decay, which is independent of $(p,q)$. At small times $0<t\ll 1$, we establish bounds on $\parallel {e^{-t{q}^w(x,D)}}_{L^p\to L^q}\parallel$ for $(p,q)$ with $1\le p\le q\le \infty$ that are polynomial in $t^{-1}$.