$L^p$-bounds for semigroups generated by non-elliptic quadratic differential operators
نویسندگان
چکیده
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 Weyl quantization of complex-valued form $q$ defined phase space $\mathbb{R}^{2n}$ with non-negative real part $\operatorname{Re} q 0$ and trivial singular space. Specifically, show $e^{-tq^w(x,D)}$ bounded from $L^p(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$ all > whenever $1 \le p \infty$, prove bounds $\parallel { e^{-tq^w(x,D)}}{L^p \rightarrow L^q} \parallel$ in both large \gg 1$ small $0 < t \ll time regimes.Regardin g $L^p L^q$ evolution at times, $\parallel{e^{-tq^w(x,D)}}{L^p exponentially decaying as determine precise rate exponential decay, which independent $(p,q)$. At times 1$, {e^{-tq^w(x,D)}}\_{L^p $(p,q)$ \infty$ are polynomial $t^{-1}$.
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ژورنال
عنوان ژورنال: Journal of spectral theory
سال: 2023
ISSN: ['1664-039X', '1664-0403']
DOI: https://doi.org/10.4171/jst/426