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

Abstract

In this note, we establish L^p -bounds for the semigroup e^{-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 \operatorname{Re} q \ge 0 and trivial singular space. Specifically, we show that e^{-tq^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^{-tq^w(x,D)}}_{L^p \rightarrow L^q} \parallel in both the large t \gg 1 and small 0 < t \ll 1 time regimes.Regardin g L^p \rightarrow L^q bounds for the evolution semigroup at large times, we show that \parallel{e^{-tq^w(x,D)}}_{L^p \rightarrow L^q} \parallel 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 \parallel {e^{-tq^w(x,D)}}_{L^p \rightarrow L^q} \parallel for (p,q) with 1 \le p \le q \le \infty that are polynomial in t^{-1} .