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

In this note, we establish $L^p$-bounds for the semigroup $e^{-tq^w(x,D)}$, $t \ge 0$, 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 $\textrm{Re} \, q \ge 0$ and trivial singular space. Specifically, we show that $e^{-tq^w(x,D)}$ is bounded $L^p(\mathbb{R}^n) \rightarrow L^q(\mathbb{R}^n)$ for all $t > 0$ whenever $1 \le p \le q \le \infty$, and we prove bounds on $||e^{-tq^w(x,D)}||_{L^p \rightarrow L^q}$ in both the large $t \gg 1$ and small $0 < t \ll 1$ time regimes. Regarding $L^p \rightarrow L^q$ bounds for the evolution semigroup at large times, we show that $||e^{-tq^w(x,D)}||_{L^p \rightarrow L^q}$ is exponentially decaying as $t \rightarrow \infty$, 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 $||e^{-tq^w(x,D)}||_{L^p \rightarrow L^q}$ for $(p,q)$ with $1 \le p \le q \le \infty$ that are polynomial in $t^{-1}$.