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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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