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Riesz transforms and the wave equation for the Hermite operator. (English) Zbl 0709.35068
Let $$A_ j=-(d/dx_ j)+x_ j$$ and $$A^*_ j=d/dx_ j+x_ j$$ be the creation and the annihilation operators and $$H=(-\Delta +| x|^ 2)=(1/2)\sum^{n}_{j=1}(A_ jA^*_ j+A^*_ jA_ j)$$ be the Hermite operator. Analogous to the classical Riesz transforms we define $$R_ j=H^{-1/2}A_ j$$ and $$R^*_ j=H^{-1/2}A^*_ j$$ and prove that they are bounded on $$L^ p({\mathbb{R}}^ n)$$, $$1<p<\infty$$. We also study the Cauchy problem for the wave equation $$\partial^ 2_ tu=Hu$$ and prove certain $$L^ p-L^ 2$$ mapping properties of the solution operators $$H^{-1/2}\sin (tH^{1/2})$$ and $$\cos (tH^{1/2})$$.
Reviewer: S.Thangavelu

##### MSC:
 35L05 Wave equation 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35L15 Initial value problems for second-order hyperbolic equations
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##### References:
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