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

2021/04/29 by White, Francis
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2104.14613

Abstract

In this note, we establish L^p-bounds for the semigroup e^-tqw(x,D), t ≥ 0, generated by a quadratic differential operator q^w(x,D) on ℝⁿ that is the Weyl quantization of a complex-valued quadratic form q defined on the phase space ℝ²ⁿ with non-negative real part \textrmRe q ≥ 0 and trivial singular space. Specifically, we show that e^-tqw(x,D) is bounded L^p(ℝⁿ) → L^q(ℝⁿ) for all t > 0 whenever 1 ≤ p ≤ q ≤ ∞, and we prove bounds on ||e^-tqw(x,D)||_{L^p → L^q} in both the large t ≫ 1 and small 0 < t ≪ 1 time regimes. Regarding L^p → L^q bounds for the evolution semigroup at large times, we show that ||e^-tqw(x,D)||_{L^p → L^q} is exponentially decaying as t → ∞, and we determine the precise rate of exponential decay, which is independent of (p,q). At small times 0 < t ≪ 1, we establish bounds on ||e^-tqw(x,D)||_{L^p → L^q} for (p,q) with 1 ≤ p ≤ q ≤ ∞ that are polynomial in t^-1.

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