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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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