Gaussian estimates and regularized groups

Zheng, Quan; Zhang, Jizhou

doi:10.1090/S0002-9939-99-04649-3

Gaussian estimates and regularized groups
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by Quan Zheng and Jizhou Zhang PDF

Proc. Amer. Math. Soc. 127 (1999), 1089-1096 Request permission

Abstract:

We show that if a bounded analytic semigroup $\{T(z)\}_{ \operatorname {Re}z>0}$ on $L^2({\boldsymbol {\Omega }} )$ $({\boldsymbol {\Omega }} \subset \mathbf {R} ^n)$ satisfies a Gaussian estimate of order $m$ and $A_p$ is the generator of its consistent semigroup on $L^p({\boldsymbol {\Omega }} )$ $(1\le p<\infty )$, then $iA_p$ generates a $(1-A_p)^{-\alpha }$-regularized group on $L^p({\boldsymbol {\Omega }} )$ where $\alpha >2n |\frac {1}{2}-\frac {1}{p}|$. We obtain the estimate of $(\lambda -A_p)^{-1}$ ($|\operatorname {arg}\lambda |<\pi$) and the $p$-independence of $\sigma (A_p)$, and give applications to Schrödinger operators and elliptic operators of higher order.

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