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\(L^p\) eigenfunction bounds for the Hermite operator. (English) Zbl 1075.35020
The authors obtain \(L^{p}\) eigenfunction bounds for the Hermite operator \(H=-\Delta+x^{2}\) in \({\mathbb R}^{n}\) and also for a larger class of related operator of the form \(H_{V}=-\Delta+V\). The main results are the following.
Theorem 1. (i) The operators \(e^{itH}\) satisfy\(\| e^{H}\|_{L^{1}\to L^{\infty}}\lesssim |\sin t|^{-n/2}.\) (ii) Let \(V\) be a potential that satisfies \(|\partial^{\alpha}V|\leq c_{\alpha}\), \(|\alpha|\geq 2.\) Then for \(|t|\ll 1\), the operators \(e^{H}_{V}\) satisfy \(\| e^{H}_{V}\|_{L^{1}\to L^{\infty}}\lesssim |t|^{-n/2}.\)
Theorem 2. (Strichartz estimate). Let \(V\) be a potential that satisfies \(|\partial^{\alpha}V|\lesssim 2,\quad |\alpha|=1.\)
Then the solution \(u\) to \((i\partial_{t}-H_{V})u=f\), \(u(0)=u_{0},\) satisfies \[ \| u\|_{L^{p_{1}}(0,1;L^{q_{1}})}\lesssim\| u_{0}\|_{L^{2}}+\| f\|_{L^{p_{2}^{\prime}}(0,1;L^{q_{2}^{\prime}})}, \] whenever the pairs \((p_{1},q_{1})\) and \((p_{2},q_{2})\) are subject to \[ \frac{2}{p}+\frac{n-1}{q}=\frac{n-1}{2},\quad2\leq p\leq\infty,\quad(n,p,q)\neq (2,2,\infty). \] Put \(D^{\text{int}}_{j}=\{|x|\in[\lambda(1-2^{-2(j-1)}),\lambda(1-2^{-2(j+1)})]\},\quad 1\leq 2^{j}\leq\lambda^{2/3}\), \(D^{\text{bd}}=\{||x|-\lambda|\leq\lambda^{-1/3}\},\) \(D^{\text{ext}}=\{|x|>\lambda+\lambda^{-1/3}/2\}\), \[ \| f\|_{l^{\infty}_{\lambda}L^{p}}^{q}=\| f\|^{q}_{L^{p}(D^{\text{ext}})}+\| f\|^{q}_{L^{p}(D^{\text{bd}})}+\sum_{1\leq j}^{2^{j}\leq\lambda^{2/3}}\| f\|^{q}_{L^{p}(D^{\text{int}}_{j})}, \]
\[ y=\lambda^{-2/3}(\lambda^{2}-x^{2}),\quad \langle y\rangle_{-}=1+y_{-},\quad \langle y\rangle_{+}=1+y_{+}. \]
Theorem 3 (Weighted \(L^{p}\) eigenfunction bound). (i) Let \(2\leq p\leq 2(n+1)/(n-1)\). Then \[ \|\lambda^{1/3-(n/3)(1/2-1/p)}\langle y\rangle_{+}^{-1/4+((n+3)/4)(1/2-a/p)}\langle y\rangle_{-}^{1-(n/2)(1/2-1/p)}\phi\|_{l^{\infty}_{\lambda}L^{p}} \lesssim \|\phi\|_{L^{2}}+\| (H-\lambda^{2})\phi\|_{L^{2}}. \]
(ii) Let \(2(n+1)/(n-1)\leq p\leq\infty\). Let \(N\) be a positive integer. Then \[ \|\lambda^{1/3-n/3(1/2-1/p)}\langle y\rangle_{+}^{1/2-(n/2)(1/2-1/p)}\langle y\rangle_{-}^{N}P_{\lambda^{2}}\phi\|_{l^{\infty}_{\lambda}L^{p}}\lesssim\|\phi\|_{L^{2}}, \] where \(P_{k}\) is a spectral projector for \(H\).
The authors get similar estimates for \(H_{V}\) with a positive potential \(V\) that satisfies \(V\sim |x|^{2}\), \(|\nabla V|\sim |x|\), and \(|\partial^{2}_{x}V|\lesssim 1\).

MSC:
35P05 General topics in linear spectral theory for PDEs
35B60 Continuation and prolongation of solutions to PDEs
35S05 Pseudodifferential operators as generalizations of partial differential operators
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